{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "# 5D Kerr-AdS spacetime with a Nambu-Goto string\n", "\n", "## Generic case (a,b) in global AdS coordinates" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This [SageMath](https://www.sagemath.org/) notebook is relative to the article *Heavy quarks in rotating plasma via holography* by Anastasia A. Golubtsova, Eric Gourgoulhon and Marina K. Usova, [arXiv:2107.11672](https://arxiv.org/abs/2107.11672).\n", "\n", "The involved differential geometry computations are based on tools developed through the [SageManifolds](https://sagemanifolds.obspm.fr) project." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "*NB:* a version of SageMath at least equal to 9.1 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.3, Release Date: 2021-05-09'" ] }, "execution_count": 1, "metadata": { }, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "First we set up the notebook to display mathematical objects using LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Since some computations are quite long, we ask for running them in parallel on 8 cores:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "Parallelism().set(nproc=8)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## Spacetime manifold\n", "\n", "We declare the Kerr-AdS spacetime as a 5-dimensional Lorentzian manifold:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "5-dimensional Lorentzian manifold M\n" ] } ], "source": [ "M = Manifold(5, 'M', r'\\mathcal{M}', structure='Lorentzian', metric_name='G')\n", "print(M)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Let us define **Boyer-Lindquist-type coordinates (rational polynomial version)** on $\\mathcal{M}$, via the method `chart()`, the argument of which is a string expressing the coordinates names, their ranges (the default is $(-\\infty,+\\infty)$) and their LaTeX symbols:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, r, {\\mu}, {\\phi}, {\\psi})\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, r, {\\mu}, {\\phi}, {\\psi})\\right)$$" ], "text/plain": [ "Chart (M, (t, r, mu, ph, ps))" ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL. = M.chart(r't r:(0,+oo) mu:(0,1):\\mu ph:(0,2*pi):\\phi ps:(0,2*pi):\\psi')\n", "BL" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The coordinate $\\mu$ is related to the standard Boyer-Lindquist coordinate $\\theta$ by\n", "$$ \\mu = \\cos\\theta$$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coordinate ranges are" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\mu} :\\ \\left( 0 , 1 \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\mu} :\\ \\left( 0 , 1 \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$$" ], "text/plain": [ "t: (-oo, +oo); r: (0, +oo); mu: (0, 1); ph: (0, 2*pi); ps: (0, 2*pi)" ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL.coord_range()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Note that contrary to the 4-dimensional case, the range of $\\mu$ is $(0,1)$ only (cf. Sec. 1.2 of [R.C. Myers, arXiv:1111.1903](https://arxiv.org/abs/1111.1903) or Sec. 2 of [G.W. Gibbons, H. Lüb, Don N. Page, C.N. Pope, J. Geom. Phys. **53**, 49 (2005)](https://doi.org/10.1016/j.geomphys.2004.05.001)). In other words, the range of $\\theta$ is $\\left(0, \\frac{\\pi}{2}\\right)$ only. " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## Metric tensor\n", "\n", "The 4 parameters $m$, $a$, $b$ and $\\ell$ of the Kerr-AdS spacetime are declared as symbolic variables, $a$ and $b$ being the two angular momentum parameters and $\\ell$ being related to the cosmological constant by $\\Lambda = - 6 \\ell^2$:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, a, b\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, a, b\\right)$$" ], "text/plain": [ "(m, a, b)" ] }, "execution_count": 7, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('m a b', domain='real')" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In this notebook, we set\n", "$$ \\ell = 1$$" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "l = 1" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "assume(a >= 0, a < 1)\n", "assume(b >= 0, b < 1)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Possible particular cases:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "#b = a\n", "#a = 0\n", "#b = 0" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Some auxiliary functions:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "keep_Delta = False # change to False to provide explicit expression for Delta_r, Xi_a, etc..." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "sig = (1 + r^2*l^2)/r^2\n", "costh2 = mu^2\n", "sinth2 = 1 - mu^2\n", "rho2 = r^2 + a^2*mu^2 + b^2*sinth2\n", "if keep_Delta:\n", " Delta_r = var('Delta_r', latex_name=r'\\Delta_r', domain='real')\n", " Delta_th = var('Delta_th', latex_name=r'\\Delta_\\theta', domain='real')\n", " if a == b:\n", " Xi_a = var('Xi', latex_name=r'\\Xi', domain='real')\n", " Xi_b = Xi_a\n", " else:\n", " Xi_a = var('Xi_a', latex_name=r'\\Xi_a', domain='real')\n", " Xi_b = var('Xi_b', latex_name=r'\\Xi_b', domain='real')\n", " #Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2\n", " Xi_a = 1 - a^2*l^2\n", " Xi_b = 1 - b^2*l^2\n", "else:\n", " Delta_r = (r^2+a^2)*(r^2+b^2)*sig - 2*m\n", " Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2\n", " Xi_a = 1 - a^2*l^2\n", " Xi_b = 1 - b^2*l^2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The metric is set by its components in the coordinate frame associated with the Boyer-Lindquist-type coordinates, which is the current manifold's default frame. These components can be deduced from\n", "Eq. (5.22) of the article [S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D **59**, 064005 (1999)](https://doi.org/10.1103/PhysRevD.59.064005) (the check of agreement with this equation is performed below):" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "G = M.metric()\n", "tmp = 1/rho2*( -Delta_r + Delta_th*(a^2*sinth2 + b^2*mu^2) + a^2*b^2*sig )\n", "G[0,0] = tmp.simplify_full()\n", "tmp = a*sinth2/(rho2*Xi_a)*( Delta_r - (r^2+a^2)*(Delta_th + b^2*sig) )\n", "G[0,3] = tmp.simplify_full()\n", "tmp = b*mu^2/(rho2*Xi_b)*( Delta_r - (r^2+b^2)*(Delta_th + a^2*sig) )\n", "G[0,4] = tmp.simplify_full()\n", "G[1,1] = (rho2/Delta_r).simplify_full()\n", "G[2,2] = (rho2/Delta_th/(1-mu^2)).simplify_full()\n", "tmp = sinth2/(rho2*Xi_a^2)*( - Delta_r*a^2*sinth2 + (r^2+a^2)^2*(Delta_th + sig*b^2*sinth2) ) \n", "G[3,3] = tmp.simplify_full()\n", "tmp = a*b*sinth2*mu^2/(rho2*Xi_a*Xi_b)*( - Delta_r + sig*(r^2+a^2)*(r^2+b^2) )\n", "G[3,4] = tmp.simplify_full()\n", "tmp = mu^2/(rho2*Xi_b^2)*( - Delta_r*b^2*mu^2 + (r^2+b^2)^2*(Delta_th + sig*a^2*mu^2) )\n", "G[4,4] = tmp.simplify_full()" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{{\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} - r^{4} - {\\left(a^{2} + 1\\right)} b^{2} - {\\left(a^{4} + b^{4} - {\\left(2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} - {\\left(a^{2} + b^{2} + 1\\right)} r^{2} + 2 \\, m}{{\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} + r^{2}} \\\\ G_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{3} b^{2} - {\\left(a^{5} - a^{3} b^{2}\\right)} {\\mu}^{4} - {\\left(a {\\mu}^{2} - a\\right)} r^{4} + {\\left(a^{5} - 2 \\, a^{3} b^{2} + 2 \\, a m\\right)} {\\mu}^{2} - {\\left(2 \\, a b^{2} {\\mu}^{2} + {\\left(a^{3} - a b^{2}\\right)} {\\mu}^{4} - a^{3} - a b^{2}\\right)} r^{2} - 2 \\, a m}{{\\left(a^{2} - 1\\right)} b^{2} + {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\mu}^{2} + {\\left(a^{2} - 1\\right)} r^{2}} \\\\ G_{ \\, t \\, {\\psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\psi}} } & = & -\\frac{b {\\mu}^{2} r^{4} + {\\left(a^{2} b^{3} - b^{5}\\right)} {\\mu}^{4} + {\\left(b^{5} - 2 \\, b m\\right)} {\\mu}^{2} + {\\left(2 \\, b^{3} {\\mu}^{2} + {\\left(a^{2} b - b^{3}\\right)} {\\mu}^{4}\\right)} r^{2}}{b^{4} - {\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} + {\\left(b^{2} - 1\\right)} r^{2} - b^{2}} \\\\ G_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & \\frac{r^{4} + {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2}\\right)} r^{2}}{r^{6} + {\\left(a^{2} + b^{2} + 1\\right)} r^{4} + a^{2} b^{2} + {\\left({\\left(a^{2} + 1\\right)} b^{2} + a^{2} - 2 \\, m\\right)} r^{2}} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & \\frac{{\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} + r^{2}}{{\\left(a^{2} - b^{2}\\right)} {\\mu}^{4} - {\\left(a^{2} - 2 \\, b^{2} + 1\\right)} {\\mu}^{2} - b^{2} + 1} \\\\ G_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & \\frac{{\\left(a^{6} - a^{4} - {\\left(a^{4} - a^{2}\\right)} b^{2} + 2 \\, a^{2} m\\right)} {\\mu}^{4} + {\\left({\\left(a^{2} - 1\\right)} {\\mu}^{2} - a^{2} + 1\\right)} r^{4} - {\\left(a^{4} - a^{2}\\right)} b^{2} + 2 \\, a^{2} m - {\\left(a^{6} - a^{4} - 2 \\, {\\left(a^{4} - a^{2}\\right)} b^{2} + 4 \\, a^{2} m\\right)} {\\mu}^{2} + {\\left(2 \\, {\\left(a^{2} - 1\\right)} b^{2} {\\mu}^{2} + {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\mu}^{4} - a^{4} - {\\left(a^{2} - 1\\right)} b^{2} + a^{2}\\right)} r^{2}}{{\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} + {\\left(a^{6} - 2 \\, a^{4} - {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} + {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} r^{2}} \\\\ G_{ \\, {\\phi} \\, {\\psi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, {\\left(a b m {\\mu}^{4} - a b m {\\mu}^{2}\\right)}}{{\\left(a^{2} - 1\\right)} b^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left({\\left(a^{2} - 1\\right)} b^{4} + a^{4} - {\\left(a^{4} - 1\\right)} b^{2} - a^{2}\\right)} {\\mu}^{2} + {\\left({\\left(a^{2} - 1\\right)} b^{2} - a^{2} + 1\\right)} r^{2}} \\\\ G_{ \\, {\\psi} \\, {\\psi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\psi}} } & = & -\\frac{{\\left(b^{2} - 1\\right)} {\\mu}^{2} r^{4} - {\\left(b^{6} - {\\left(a^{2} + 1\\right)} b^{4} + a^{2} b^{2} + 2 \\, b^{2} m\\right)} {\\mu}^{4} + {\\left(b^{6} - b^{4}\\right)} {\\mu}^{2} - {\\left({\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{4} - 2 \\, {\\left(b^{4} - b^{2}\\right)} {\\mu}^{2}\\right)} r^{2}}{b^{6} - 2 \\, b^{4} - {\\left(b^{6} - {\\left(a^{2} + 2\\right)} b^{4} + {\\left(2 \\, a^{2} + 1\\right)} b^{2} - a^{2}\\right)} {\\mu}^{2} + {\\left(b^{4} - 2 \\, b^{2} + 1\\right)} r^{2} + b^{2}} \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{{\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} - r^{4} - {\\left(a^{2} + 1\\right)} b^{2} - {\\left(a^{4} + b^{4} - {\\left(2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} - {\\left(a^{2} + b^{2} + 1\\right)} r^{2} + 2 \\, m}{{\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} + r^{2}} \\\\ G_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{3} b^{2} - {\\left(a^{5} - a^{3} b^{2}\\right)} {\\mu}^{4} - {\\left(a {\\mu}^{2} - a\\right)} r^{4} + {\\left(a^{5} - 2 \\, a^{3} b^{2} + 2 \\, a m\\right)} {\\mu}^{2} - {\\left(2 \\, a b^{2} {\\mu}^{2} + {\\left(a^{3} - a b^{2}\\right)} {\\mu}^{4} - a^{3} - a b^{2}\\right)} r^{2} - 2 \\, a m}{{\\left(a^{2} - 1\\right)} b^{2} + {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\mu}^{2} + {\\left(a^{2} - 1\\right)} r^{2}} \\\\ G_{ \\, t \\, {\\psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\psi}} } & = & -\\frac{b {\\mu}^{2} r^{4} + {\\left(a^{2} b^{3} - b^{5}\\right)} {\\mu}^{4} + {\\left(b^{5} - 2 \\, b m\\right)} {\\mu}^{2} + {\\left(2 \\, b^{3} {\\mu}^{2} + {\\left(a^{2} b - b^{3}\\right)} {\\mu}^{4}\\right)} r^{2}}{b^{4} - {\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} + {\\left(b^{2} - 1\\right)} r^{2} - b^{2}} \\\\ G_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & \\frac{r^{4} + {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2}\\right)} r^{2}}{r^{6} + {\\left(a^{2} + b^{2} + 1\\right)} r^{4} + a^{2} b^{2} + {\\left({\\left(a^{2} + 1\\right)} b^{2} + a^{2} - 2 \\, m\\right)} r^{2}} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & \\frac{{\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} + r^{2}}{{\\left(a^{2} - b^{2}\\right)} {\\mu}^{4} - {\\left(a^{2} - 2 \\, b^{2} + 1\\right)} {\\mu}^{2} - b^{2} + 1} \\\\ G_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & \\frac{{\\left(a^{6} - a^{4} - {\\left(a^{4} - a^{2}\\right)} b^{2} + 2 \\, a^{2} m\\right)} {\\mu}^{4} + {\\left({\\left(a^{2} - 1\\right)} {\\mu}^{2} - a^{2} + 1\\right)} r^{4} - {\\left(a^{4} - a^{2}\\right)} b^{2} + 2 \\, a^{2} m - {\\left(a^{6} - a^{4} - 2 \\, {\\left(a^{4} - a^{2}\\right)} b^{2} + 4 \\, a^{2} m\\right)} {\\mu}^{2} + {\\left(2 \\, {\\left(a^{2} - 1\\right)} b^{2} {\\mu}^{2} + {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\mu}^{4} - a^{4} - {\\left(a^{2} - 1\\right)} b^{2} + a^{2}\\right)} r^{2}}{{\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} + {\\left(a^{6} - 2 \\, a^{4} - {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} + {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} r^{2}} \\\\ G_{ \\, {\\phi} \\, {\\psi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, {\\left(a b m {\\mu}^{4} - a b m {\\mu}^{2}\\right)}}{{\\left(a^{2} - 1\\right)} b^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left({\\left(a^{2} - 1\\right)} b^{4} + a^{4} - {\\left(a^{4} - 1\\right)} b^{2} - a^{2}\\right)} {\\mu}^{2} + {\\left({\\left(a^{2} - 1\\right)} b^{2} - a^{2} + 1\\right)} r^{2}} \\\\ G_{ \\, {\\psi} \\, {\\psi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\psi}} } & = & -\\frac{{\\left(b^{2} - 1\\right)} {\\mu}^{2} r^{4} - {\\left(b^{6} - {\\left(a^{2} + 1\\right)} b^{4} + a^{2} b^{2} + 2 \\, b^{2} m\\right)} {\\mu}^{4} + {\\left(b^{6} - b^{4}\\right)} {\\mu}^{2} - {\\left({\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{4} - 2 \\, {\\left(b^{4} - b^{2}\\right)} {\\mu}^{2}\\right)} r^{2}}{b^{6} - 2 \\, b^{4} - {\\left(b^{6} - {\\left(a^{2} + 2\\right)} b^{4} + {\\left(2 \\, a^{2} + 1\\right)} b^{2} - a^{2}\\right)} {\\mu}^{2} + {\\left(b^{4} - 2 \\, b^{2} + 1\\right)} r^{2} + b^{2}} \\end{array}$$" ], "text/plain": [ "G_t,t = ((a^4 - 2*a^2*b^2 + b^4)*mu^4 - r^4 - (a^2 + 1)*b^2 - (a^4 + b^4 - (2*a^2 + 1)*b^2 + a^2)*mu^2 - (a^2 + b^2 + 1)*r^2 + 2*m)/((a^2 - b^2)*mu^2 + b^2 + r^2) \n", "G_t,ph = -(a^3*b^2 - (a^5 - a^3*b^2)*mu^4 - (a*mu^2 - a)*r^4 + (a^5 - 2*a^3*b^2 + 2*a*m)*mu^2 - (2*a*b^2*mu^2 + (a^3 - a*b^2)*mu^4 - a^3 - a*b^2)*r^2 - 2*a*m)/((a^2 - 1)*b^2 + (a^4 - (a^2 - 1)*b^2 - a^2)*mu^2 + (a^2 - 1)*r^2) \n", "G_t,ps = -(b*mu^2*r^4 + (a^2*b^3 - b^5)*mu^4 + (b^5 - 2*b*m)*mu^2 + (2*b^3*mu^2 + (a^2*b - b^3)*mu^4)*r^2)/(b^4 - (b^4 - (a^2 + 1)*b^2 + a^2)*mu^2 + (b^2 - 1)*r^2 - b^2) \n", "G_r,r = (r^4 + ((a^2 - b^2)*mu^2 + b^2)*r^2)/(r^6 + (a^2 + b^2 + 1)*r^4 + a^2*b^2 + ((a^2 + 1)*b^2 + a^2 - 2*m)*r^2) \n", "G_mu,mu = ((a^2 - b^2)*mu^2 + b^2 + r^2)/((a^2 - b^2)*mu^4 - (a^2 - 2*b^2 + 1)*mu^2 - b^2 + 1) \n", "G_ph,ph = ((a^6 - a^4 - (a^4 - a^2)*b^2 + 2*a^2*m)*mu^4 + ((a^2 - 1)*mu^2 - a^2 + 1)*r^4 - (a^4 - a^2)*b^2 + 2*a^2*m - (a^6 - a^4 - 2*(a^4 - a^2)*b^2 + 4*a^2*m)*mu^2 + (2*(a^2 - 1)*b^2*mu^2 + (a^4 - (a^2 - 1)*b^2 - a^2)*mu^4 - a^4 - (a^2 - 1)*b^2 + a^2)*r^2)/((a^4 - 2*a^2 + 1)*b^2 + (a^6 - 2*a^4 - (a^4 - 2*a^2 + 1)*b^2 + a^2)*mu^2 + (a^4 - 2*a^2 + 1)*r^2) \n", "G_ph,ps = -2*(a*b*m*mu^4 - a*b*m*mu^2)/((a^2 - 1)*b^4 - (a^2 - 1)*b^2 - ((a^2 - 1)*b^4 + a^4 - (a^4 - 1)*b^2 - a^2)*mu^2 + ((a^2 - 1)*b^2 - a^2 + 1)*r^2) \n", "G_ps,ps = -((b^2 - 1)*mu^2*r^4 - (b^6 - (a^2 + 1)*b^4 + a^2*b^2 + 2*b^2*m)*mu^4 + (b^6 - b^4)*mu^2 - ((b^4 - (a^2 + 1)*b^2 + a^2)*mu^4 - 2*(b^4 - b^2)*mu^2)*r^2)/(b^6 - 2*b^4 - (b^6 - (a^2 + 2)*b^4 + (2*a^2 + 1)*b^2 - a^2)*mu^2 + (b^4 - 2*b^2 + 1)*r^2 + b^2) " ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (2.9)\n", "\n", "We need the 1-forms $\\mathrm{d}t$, $\\mathrm{d}r$, $\\mathrm{d}\\mu$, $\\mathrm{d}\\phi$ and $\\mathrm{d}\\psi$:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathrm{d} t, \\mathrm{d} r, \\mathrm{d} {\\mu}, \\mathrm{d} {\\phi}, \\mathrm{d} {\\psi}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathrm{d} t, \\mathrm{d} r, \\mathrm{d} {\\mu}, \\mathrm{d} {\\phi}, \\mathrm{d} {\\psi}\\right)$$" ], "text/plain": [ "(1-form dt on the 5-dimensional Lorentzian manifold M,\n", " 1-form dr on the 5-dimensional Lorentzian manifold M,\n", " 1-form dmu on the 5-dimensional Lorentzian manifold M,\n", " 1-form dph on the 5-dimensional Lorentzian manifold M,\n", " 1-form dps on the 5-dimensional Lorentzian manifold M)" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" } ], "source": [ "dt, dr, dmu, dph, dps = (BL.coframe()[i] for i in M.irange())\n", "dt, dr, dmu, dph, dps" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form dt on the 5-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(dt)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In agreement with $\\mu = \\cos\\theta$, we introduce the 1-form\n", "$$\\mathrm{d}\\theta = - \\mathrm{d}\\mu /\\sin\\theta ,$$\n", "with\n", "$\\sin\\theta = \\sqrt{1-\\mu^2}$ since $\\theta\\in\\left(0, \\frac{\\pi}{2}\\right)$ :" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "dth = - 1/sqrt(1 - mu^2)*dmu" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} t + \\left( -\\frac{a {\\mu}^{2} - a}{a^{2} - 1} \\right) \\mathrm{d} {\\phi} + \\left( \\frac{b {\\mu}^{2}}{b^{2} - 1} \\right) \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} t + \\left( -\\frac{a {\\mu}^{2} - a}{a^{2} - 1} \\right) \\mathrm{d} {\\phi} + \\left( \\frac{b {\\mu}^{2}}{b^{2} - 1} \\right) \\mathrm{d} {\\psi}$$" ], "text/plain": [ "dt - (a*mu^2 - a)/(a^2 - 1) dph + b*mu^2/(b^2 - 1) dps" ] }, "execution_count": 18, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = dt - a*sinth2/Xi_a*dph - b*costh2/Xi_b*dps\n", "s1.display()" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\mathrm{d} t + \\left( \\frac{a^{2} + r^{2}}{a^{2} - 1} \\right) \\mathrm{d} {\\phi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\mathrm{d} t + \\left( \\frac{a^{2} + r^{2}}{a^{2} - 1} \\right) \\mathrm{d} {\\phi}$$" ], "text/plain": [ "a dt + (a^2 + r^2)/(a^2 - 1) dph" ] }, "execution_count": 19, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = a*dt - (r^2 + a^2)/Xi_a*dph\n", "s2.display()" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}b \\mathrm{d} t + \\left( \\frac{b^{2} + r^{2}}{b^{2} - 1} \\right) \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}b \\mathrm{d} t + \\left( \\frac{b^{2} + r^{2}}{b^{2} - 1} \\right) \\mathrm{d} {\\psi}$$" ], "text/plain": [ "b dt + (b^2 + r^2)/(b^2 - 1) dps" ] }, "execution_count": 20, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s3 = b*dt - (r^2 + b^2)/Xi_b*dps\n", "s3.display()" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a b \\mathrm{d} t + \\left( -\\frac{a^{2} b {\\mu}^{2} - a^{2} b + {\\left(b {\\mu}^{2} - b\\right)} r^{2}}{a^{2} - 1} \\right) \\mathrm{d} {\\phi} + \\left( \\frac{a b^{2} {\\mu}^{2} + a {\\mu}^{2} r^{2}}{b^{2} - 1} \\right) \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a b \\mathrm{d} t + \\left( -\\frac{a^{2} b {\\mu}^{2} - a^{2} b + {\\left(b {\\mu}^{2} - b\\right)} r^{2}}{a^{2} - 1} \\right) \\mathrm{d} {\\phi} + \\left( \\frac{a b^{2} {\\mu}^{2} + a {\\mu}^{2} r^{2}}{b^{2} - 1} \\right) \\mathrm{d} {\\psi}$$" ], "text/plain": [ "a*b dt - (a^2*b*mu^2 - a^2*b + (b*mu^2 - b)*r^2)/(a^2 - 1) dph + (a*b^2*mu^2 + a*mu^2*r^2)/(b^2 - 1) dps" ] }, "execution_count": 21, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s4 = a*b*dt - b*(r^2 + a^2)*sinth2/Xi_a * dph - a*(r^2 + b^2)*costh2/Xi_b * dps\n", "s4.display()" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "G0 = - Delta_r/rho2 * s1*s1 + Delta_th*sinth2/rho2 * s2*s2 \\\n", " + Delta_th*costh2/rho2 * s3*s3 + rho2/Delta_r * dr*dr \\\n", " + rho2/Delta_th * dth*dth + sig/rho2 * s4*s4" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check of Eq. (2.9):" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 23, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G0 == G" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## Einstein equation\n", "\n", "The Ricci tensor of $g$ is" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "if not keep_Delta:\n", " # Ric = G.ricci()\n", " # print(Ric)\n", " pass" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "if not keep_Delta:\n", " # show(Ric.display_comp(only_nonredundant=True))\n", " pass" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Let us check that $g$ is a solution of the vacuum Einstein equation with the cosmological constant $\\Lambda = - 6 \\ell^2$:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "Lambda = -6*l^2\n", "if not keep_Delta:\n", " # print(Ric == 2/3*Lambda*G)\n", " pass" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Conformal metric at the boundary $r\\to +\\infty$ (check of Eq. (2.11))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The conformal metric:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "H = G / (1 + r^2)\n", "H.set_name('H')" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{r^{4} + {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2}\\right)} r^{2}}{{\\left(r^{6} + {\\left(a^{2} + b^{2} + 1\\right)} r^{4} + a^{2} b^{2} + {\\left({\\left(a^{2} + 1\\right)} b^{2} + a^{2} - 2 \\, m\\right)} r^{2}\\right)} {\\left(r^{2} + 1\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{r^{4} + {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2}\\right)} r^{2}}{{\\left(r^{6} + {\\left(a^{2} + b^{2} + 1\\right)} r^{4} + a^{2} b^{2} + {\\left({\\left(a^{2} + 1\\right)} b^{2} + a^{2} - 2 \\, m\\right)} r^{2}\\right)} {\\left(r^{2} + 1\\right)}}$$" ], "text/plain": [ "(r^4 + ((a^2 - b^2)*mu^2 + b^2)*r^2)/((r^6 + (a^2 + b^2 + 1)*r^4 + a^2*b^2 + ((a^2 + 1)*b^2 + a^2 - 2*m)*r^2)*(r^2 + 1))" ] }, "execution_count": 28, "metadata": { }, "output_type": "execute_result" } ], "source": [ "H[1,1]" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us introduce a function to perform asymptotic expansions up to a given order:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "u = var('u')\n", "def asympt(xx, rr, order):\n", " r\"\"\"\n", " Expansion in terms of 1/rr\n", "\n", " INPUT:\n", "\n", " - ``xx`` -- symbolic expression to expand\n", " - ``rr`` -- symbolic variable, the inverse of which is the expansion small parameter\n", " - ``order`` -- order of the expansion\n", "\n", " OUTPUT:\n", "\n", " - symbolic expression representing ``xx`` truncated at degree ``order`` in terms\n", " of ``1/rr``\n", "\n", " \"\"\"\n", " xx = xx.subs({rr: 1/u}).simplify_full()\n", " xx = xx.series(u, order+1).truncate().simplify_full()\n", " xx = xx.subs({u: 1/rr}).simplify_full()\n", " return xx " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Expansion to order $1/r^0$ provides the conformal metric on the boundary $r\\to +\\infty$:" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}H_0 = -\\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{a {\\mu}^{2} - a}{a^{2} - 1} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{b {\\mu}^{2}}{b^{2} - 1} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\psi} + \\left( \\frac{1}{{\\left(a^{2} - b^{2}\\right)} {\\mu}^{4} - {\\left(a^{2} - 2 \\, b^{2} + 1\\right)} {\\mu}^{2} - b^{2} + 1} \\right) \\mathrm{d} {\\mu}\\otimes \\mathrm{d} {\\mu} + \\left( \\frac{a {\\mu}^{2} - a}{a^{2} - 1} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( \\frac{{\\mu}^{2} - 1}{a^{2} - 1} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{b {\\mu}^{2}}{b^{2} - 1} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} t + \\left( -\\frac{{\\mu}^{2}}{b^{2} - 1} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}H_0 = -\\mathrm{d} t\\otimes \\mathrm{d} t + \\left( \\frac{a {\\mu}^{2} - a}{a^{2} - 1} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{b {\\mu}^{2}}{b^{2} - 1} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\psi} + \\left( \\frac{1}{{\\left(a^{2} - b^{2}\\right)} {\\mu}^{4} - {\\left(a^{2} - 2 \\, b^{2} + 1\\right)} {\\mu}^{2} - b^{2} + 1} \\right) \\mathrm{d} {\\mu}\\otimes \\mathrm{d} {\\mu} + \\left( \\frac{a {\\mu}^{2} - a}{a^{2} - 1} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( \\frac{{\\mu}^{2} - 1}{a^{2} - 1} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{b {\\mu}^{2}}{b^{2} - 1} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} t + \\left( -\\frac{{\\mu}^{2}}{b^{2} - 1} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} {\\psi}$$" ], "text/plain": [ "H_0 = -dt*dt + (a*mu^2 - a)/(a^2 - 1) dt*dph - b*mu^2/(b^2 - 1) dt*dps + 1/((a^2 - b^2)*mu^4 - (a^2 - 2*b^2 + 1)*mu^2 - b^2 + 1) dmu*dmu + (a*mu^2 - a)/(a^2 - 1) dph*dt + (mu^2 - 1)/(a^2 - 1) dph*dph - b*mu^2/(b^2 - 1) dps*dt - mu^2/(b^2 - 1) dps*dps" ] }, "execution_count": 30, "metadata": { }, "output_type": "execute_result" } ], "source": [ "H0 = M.sym_bilin_form_field(name='H_0')\n", "for i in M.irange():\n", " for j in M.irange(i):\n", " H0[i, j] = asympt(H[i,j].expr(), r, 0)\n", "H0.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This agrees with Eq. (2.11); in particular, we have" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\left(a^{2} {\\mu}^{2} - b^{2} {\\mu}^{2} + b^{2} - 1\\right)} {\\left({\\mu} + 1\\right)} {\\left({\\mu} - 1\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\left(a^{2} {\\mu}^{2} - b^{2} {\\mu}^{2} + b^{2} - 1\\right)} {\\left({\\mu} + 1\\right)} {\\left({\\mu} - 1\\right)}}$$" ], "text/plain": [ "1/((a^2*mu^2 - b^2*mu^2 + b^2 - 1)*(mu + 1)*(mu - 1))" ] }, "execution_count": 31, "metadata": { }, "output_type": "execute_result" } ], "source": [ "H0[2,2].factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Global AdS coordinates" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(T, y, {\\chi}, {\\Phi}, {\\Psi})\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(T, y, {\\chi}, {\\Phi}, {\\Psi})\\right)$$" ], "text/plain": [ "Chart (M, (T, y, ch, Ph, Ps))" ] }, "execution_count": 32, "metadata": { }, "output_type": "execute_result" } ], "source": [ "ADS. = M.chart(r'T y:(0,+oo) ch:(0,1):\\chi Ph:(0,2*pi):\\Phi Ps:(0,2*pi):\\Psi')\n", "ADS" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}T :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\chi} :\\ \\left( 0 , 1 \\right) ;\\quad {\\Phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\Psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}T :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\chi} :\\ \\left( 0 , 1 \\right) ;\\quad {\\Phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\Psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$$" ], "text/plain": [ "T: (-oo, +oo); y: (0, +oo); ch: (0, 1); Ph: (0, 2*pi); Ps: (0, 2*pi)" ] }, "execution_count": 33, "metadata": { }, "output_type": "execute_result" } ], "source": [ "ADS.coord_range()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The transition from the Boyer-Lindquist coordinates to the global AdS coordinates is derived from Eq. (5.24) of [S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D **59**, 064005 (1999)](https://doi.org/10.1103/PhysRevD.59.064005):" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The following assumptions are required to perform simplifications:" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "if not (a == 0 or b == 0 or b == a):\n", " assume((a^2 - b^2)*mu^2 + b^2 + r^2 > 0)\n", " assume((a^2 - b^2)*ch^2 - a^2 + 1 > 0)" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\verb|t|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|r|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, r > 0, \\verb|mu|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\mu} > 0, {\\mu} < 1, \\verb|ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\phi} > 0, {\\phi} < 2 \\, \\pi, \\verb|ps|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\psi} > 0, {\\psi} < 2 \\, \\pi, \\verb|m|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|a|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|b|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, a \\geq 0, a < 1, b \\geq 0, b < 1, \\verb|T|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|y|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, y > 0, \\verb|ch|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\chi} > 0, {\\chi} < 1, \\verb|Ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\Phi} > 0, {\\Phi} < 2 \\, \\pi, \\verb|Ps|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\Psi} > 0, {\\Psi} < 2 \\, \\pi, {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} + r^{2} > 0, {\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1 > 0\\right]$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\verb|t|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|r|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, r > 0, \\verb|mu|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\mu} > 0, {\\mu} < 1, \\verb|ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\phi} > 0, {\\phi} < 2 \\, \\pi, \\verb|ps|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\psi} > 0, {\\psi} < 2 \\, \\pi, \\verb|m|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|a|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|b|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, a \\geq 0, a < 1, b \\geq 0, b < 1, \\verb|T|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|y|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, y > 0, \\verb|ch|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\chi} > 0, {\\chi} < 1, \\verb|Ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\Phi} > 0, {\\Phi} < 2 \\, \\pi, \\verb|Ps|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\Psi} > 0, {\\Psi} < 2 \\, \\pi, {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} + r^{2} > 0, {\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1 > 0\\right]$$" ], "text/plain": [ "[t is real,\n", " r is real,\n", " r > 0,\n", " mu is real,\n", " mu > 0,\n", " mu < 1,\n", " ph is real,\n", " ph > 0,\n", " ph < 2*pi,\n", " ps is real,\n", " ps > 0,\n", " ps < 2*pi,\n", " m is real,\n", " a is real,\n", " b is real,\n", " a >= 0,\n", " a < 1,\n", " b >= 0,\n", " b < 1,\n", " T is real,\n", " y is real,\n", " y > 0,\n", " ch is real,\n", " ch > 0,\n", " ch < 1,\n", " Ph is real,\n", " Ph > 0,\n", " Ph < 2*pi,\n", " Ps is real,\n", " Ps > 0,\n", " Ps < 2*pi,\n", " (a^2 - b^2)*mu^2 + b^2 + r^2 > 0,\n", " (a^2 - b^2)*ch^2 - a^2 + 1 > 0]" ] }, "execution_count": 35, "metadata": { }, "output_type": "execute_result" } ], "source": [ "assumptions()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We import the function `simplify_sqrt_real` to simplify some square roots, which would not be simplified with `simplify_full`:" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "from sage.manifolds.utilities import simplify_sqrt_real" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "ys = sqrt(Xi_b*(r^2 + a^2)*(1-mu^2) + Xi_a*(r^2 + b^2)*mu^2) \\\n", " /(sqrt(Xi_a)*sqrt(Xi_b))\n", "ys = simplify_sqrt_real(ys.simplify_full())\n", "\n", "chs = sqrt(Xi_a)*sqrt(r^2 + b^2)*mu / sqrt(Xi_a*(r^2 + b^2)*mu^2 \n", " + Xi_b*(r^2 + a^2)*(1 - mu^2))\n", "chs = simplify_sqrt_real(chs.simplify_full())" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}}}{\\sqrt{a + 1} \\sqrt{-a + 1} \\sqrt{b + 1} \\sqrt{-b + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}}}{\\sqrt{a + 1} \\sqrt{-a + 1} \\sqrt{b + 1} \\sqrt{-b + 1}}$$" ], "text/plain": [ "sqrt(-a^2*b^2 - (a^2 - b^2)*mu^2 - ((a^2 - b^2)*mu^2 + b^2 - 1)*r^2 + a^2)/(sqrt(a + 1)*sqrt(-a + 1)*sqrt(b + 1)*sqrt(-b + 1))" ] }, "execution_count": 38, "metadata": { }, "output_type": "execute_result" } ], "source": [ "ys" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{b^{2} + r^{2}} \\sqrt{a + 1} \\sqrt{-a + 1} {\\mu}}{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{b^{2} + r^{2}} \\sqrt{a + 1} \\sqrt{-a + 1} {\\mu}}{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}}}$$" ], "text/plain": [ "sqrt(b^2 + r^2)*sqrt(a + 1)*sqrt(-a + 1)*mu/sqrt(-a^2*b^2 - (a^2 - b^2)*mu^2 - ((a^2 - b^2)*mu^2 + b^2 - 1)*r^2 + a^2)" ] }, "execution_count": 39, "metadata": { }, "output_type": "execute_result" } ], "source": [ "chs" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} T & = & t \\\\ y & = & \\frac{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}}}{\\sqrt{a + 1} \\sqrt{-a + 1} \\sqrt{b + 1} \\sqrt{-b + 1}} \\\\ {\\chi} & = & \\frac{\\sqrt{b^{2} + r^{2}} \\sqrt{a + 1} \\sqrt{-a + 1} {\\mu}}{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}}} \\\\ {\\Phi} & = & a t + {\\phi} \\\\ {\\Psi} & = & b t + {\\psi} \\end{array}\\right.$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} T & = & t \\\\ y & = & \\frac{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}}}{\\sqrt{a + 1} \\sqrt{-a + 1} \\sqrt{b + 1} \\sqrt{-b + 1}} \\\\ {\\chi} & = & \\frac{\\sqrt{b^{2} + r^{2}} \\sqrt{a + 1} \\sqrt{-a + 1} {\\mu}}{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}}} \\\\ {\\Phi} & = & a t + {\\phi} \\\\ {\\Psi} & = & b t + {\\psi} \\end{array}\\right.$$" ], "text/plain": [ "T = t\n", "y = sqrt(-a^2*b^2 - (a^2 - b^2)*mu^2 - ((a^2 - b^2)*mu^2 + b^2 - 1)*r^2 + a^2)/(sqrt(a + 1)*sqrt(-a + 1)*sqrt(b + 1)*sqrt(-b + 1))\n", "ch = sqrt(b^2 + r^2)*sqrt(a + 1)*sqrt(-a + 1)*mu/sqrt(-a^2*b^2 - (a^2 - b^2)*mu^2 - ((a^2 - b^2)*mu^2 + b^2 - 1)*r^2 + a^2)\n", "Ph = a*t + ph\n", "Ps = b*t + ps" ] }, "execution_count": 40, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL_to_ADS = BL.transition_map(ADS, [t, ys, chs, ph + a*t, ps + b*t])\n", "BL_to_ADS.display()" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}$$" ], "text/plain": [ "((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2" ] }, "execution_count": 41, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Discr = (a^2 + b^2 + y^2*((b^2 - a^2)*ch^2 + a^2 - 1))^2 - 4*a^2*b^2 \\\n", " + 4*y^2*((a^2 - b^2)*ch^2 + b^2*(1 - a^2))\n", "Discr = Discr.simplify_full()\n", "Discr" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "sqrDiscr = simplify_sqrt_real(sqrt(Discr))" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, {\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - \\frac{1}{2} \\, a^{2} - \\frac{1}{2} \\, b^{2} + \\frac{1}{2} \\, \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, {\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - \\frac{1}{2} \\, a^{2} - \\frac{1}{2} \\, b^{2} + \\frac{1}{2} \\, \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}$$" ], "text/plain": [ "1/2*((a^2 - b^2)*ch^2 - a^2 + 1)*y^2 - 1/2*a^2 - 1/2*b^2 + 1/2*sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2)" ] }, "execution_count": 43, "metadata": { }, "output_type": "execute_result" } ], "source": [ "rs2 = 1/2*(y^2*((a^2 - b^2)*ch^2 + 1 - a^2) - a^2 - b^2 + sqrDiscr)\n", "rs2 = rs2.simplify_full()\n", "rs2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check:" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$$" ], "text/plain": [ "0 == 0" ] }, "execution_count": 44, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = (rs2 + a^2)*(rs2 + b^2) - (rs2 + a^2)*(1 - b^2)*y^2*ch^2 \\\n", " - (rs2 + b^2)*(1 - a^2)*y^2*(1 - ch^2) == 0\n", "s.simplify_full()" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "rs = simplify_sqrt_real(sqrt(rs2))" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{b + 1} \\sqrt{-b + 1} {\\chi} y}{\\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} + b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{b + 1} \\sqrt{-b + 1} {\\chi} y}{\\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} + b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}}}$$" ], "text/plain": [ "sqrt(2)*sqrt(b + 1)*sqrt(-b + 1)*ch*y/sqrt(((a^2 - b^2)*ch^2 - a^2 + 1)*y^2 - a^2 + b^2 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2))" ] }, "execution_count": 46, "metadata": { }, "output_type": "execute_result" } ], "source": [ "mus = sqrt(1 - b^2)/sqrt(rs2 + b^2)*y*ch\n", "mus = simplify_sqrt_real(mus.simplify_full())\n", "mus" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " t == t *passed*\n", " r == r *passed*\n", " mu == mu *passed*\n", " ph == ph *passed*\n", " ps == ps *passed*\n", " T == T *passed*\n", " y == y *passed*\n", " ch == ch *passed*\n", " Ph == Ph *passed*\n", " Ps == Ps *passed*\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & T \\\\ r & = & \\frac{1}{2} \\, \\sqrt{2} \\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} - b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}} \\\\ {\\mu} & = & \\frac{\\sqrt{2} \\sqrt{b + 1} \\sqrt{-b + 1} {\\chi} y}{\\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} + b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}}} \\\\ {\\phi} & = & -T a + {\\Phi} \\\\ {\\psi} & = & -T b + {\\Psi} \\end{array}\\right.$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & T \\\\ r & = & \\frac{1}{2} \\, \\sqrt{2} \\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} - b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}} \\\\ {\\mu} & = & \\frac{\\sqrt{2} \\sqrt{b + 1} \\sqrt{-b + 1} {\\chi} y}{\\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} + b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}}} \\\\ {\\phi} & = & -T a + {\\Phi} \\\\ {\\psi} & = & -T b + {\\Psi} \\end{array}\\right.$$" ], "text/plain": [ "t = T\n", "r = 1/2*sqrt(2)*sqrt(((a^2 - b^2)*ch^2 - a^2 + 1)*y^2 - a^2 - b^2 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2))\n", "mu = sqrt(2)*sqrt(b + 1)*sqrt(-b + 1)*ch*y/sqrt(((a^2 - b^2)*ch^2 - a^2 + 1)*y^2 - a^2 + b^2 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2))\n", "ph = -T*a + Ph\n", "ps = -T*b + Ps" ] }, "execution_count": 47, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL_to_ADS.set_inverse(T, rs, mus, Ph - a*T, Ps - b*T, \n", " verbose=True)\n", "BL_to_ADS.inverse().display()" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n", "1 & 0 & 0 & 0 & 0 \\\\\n", "0 & -\\frac{{\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r}{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}} \\sqrt{a + 1} \\sqrt{-a + 1} \\sqrt{b + 1} \\sqrt{-b + 1}} & -\\frac{{\\left(a^{2} - b^{2}\\right)} {\\mu} r^{2} + {\\left(a^{2} - b^{2}\\right)} {\\mu}}{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}} \\sqrt{a + 1} \\sqrt{-a + 1} \\sqrt{b + 1} \\sqrt{-b + 1}} & 0 & 0 \\\\\n", "0 & -\\frac{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}} {\\left({\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{3} - {\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}\\right)} \\sqrt{a + 1} \\sqrt{-a + 1} r}{{\\left(a^{4} b^{4} - 2 \\, a^{4} b^{2} + {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} + {\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} + b^{4} - 2 \\, {\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} - 2 \\, b^{2} + 1\\right)} r^{4} + a^{4} - 2 \\, {\\left(a^{2} b^{4} + a^{4} - {\\left(a^{4} + a^{2}\\right)} b^{2}\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{2} b^{4} + {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} - 2 \\, a^{2} b^{2} - {\\left({\\left(a^{2} + 1\\right)} b^{4} + a^{4} - {\\left(a^{4} + 2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} + a^{2}\\right)} r^{2}\\right)} \\sqrt{b^{2} + r^{2}}} & -\\frac{{\\left(a^{2} b^{2} + {\\left(b^{2} - 1\\right)} r^{2} - a^{2}\\right)} \\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}} \\sqrt{b^{2} + r^{2}} \\sqrt{a + 1} \\sqrt{-a + 1}}{a^{4} b^{4} - 2 \\, a^{4} b^{2} + {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} + {\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} + b^{4} - 2 \\, {\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} - 2 \\, b^{2} + 1\\right)} r^{4} + a^{4} - 2 \\, {\\left(a^{2} b^{4} + a^{4} - {\\left(a^{4} + a^{2}\\right)} b^{2}\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{2} b^{4} + {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} - 2 \\, a^{2} b^{2} - {\\left({\\left(a^{2} + 1\\right)} b^{4} + a^{4} - {\\left(a^{4} + 2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} + a^{2}\\right)} r^{2}} & 0 & 0 \\\\\n", "a & 0 & 0 & 1 & 0 \\\\\n", "b & 0 & 0 & 0 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n", "1 & 0 & 0 & 0 & 0 \\\\\n", "0 & -\\frac{{\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r}{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}} \\sqrt{a + 1} \\sqrt{-a + 1} \\sqrt{b + 1} \\sqrt{-b + 1}} & -\\frac{{\\left(a^{2} - b^{2}\\right)} {\\mu} r^{2} + {\\left(a^{2} - b^{2}\\right)} {\\mu}}{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}} \\sqrt{a + 1} \\sqrt{-a + 1} \\sqrt{b + 1} \\sqrt{-b + 1}} & 0 & 0 \\\\\n", "0 & -\\frac{\\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}} {\\left({\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{3} - {\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}\\right)} \\sqrt{a + 1} \\sqrt{-a + 1} r}{{\\left(a^{4} b^{4} - 2 \\, a^{4} b^{2} + {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} + {\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} + b^{4} - 2 \\, {\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} - 2 \\, b^{2} + 1\\right)} r^{4} + a^{4} - 2 \\, {\\left(a^{2} b^{4} + a^{4} - {\\left(a^{4} + a^{2}\\right)} b^{2}\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{2} b^{4} + {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} - 2 \\, a^{2} b^{2} - {\\left({\\left(a^{2} + 1\\right)} b^{4} + a^{4} - {\\left(a^{4} + 2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} + a^{2}\\right)} r^{2}\\right)} \\sqrt{b^{2} + r^{2}}} & -\\frac{{\\left(a^{2} b^{2} + {\\left(b^{2} - 1\\right)} r^{2} - a^{2}\\right)} \\sqrt{-a^{2} b^{2} - {\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} - {\\left({\\left(a^{2} - b^{2}\\right)} {\\mu}^{2} + b^{2} - 1\\right)} r^{2} + a^{2}} \\sqrt{b^{2} + r^{2}} \\sqrt{a + 1} \\sqrt{-a + 1}}{a^{4} b^{4} - 2 \\, a^{4} b^{2} + {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} + {\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} + b^{4} - 2 \\, {\\left(b^{4} - {\\left(a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} - 2 \\, b^{2} + 1\\right)} r^{4} + a^{4} - 2 \\, {\\left(a^{2} b^{4} + a^{4} - {\\left(a^{4} + a^{2}\\right)} b^{2}\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{2} b^{4} + {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\mu}^{4} - 2 \\, a^{2} b^{2} - {\\left({\\left(a^{2} + 1\\right)} b^{4} + a^{4} - {\\left(a^{4} + 2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\mu}^{2} + a^{2}\\right)} r^{2}} & 0 & 0 \\\\\n", "a & 0 & 0 & 1 & 0 \\\\\n", "b & 0 & 0 & 0 & 1\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ 1 0 0 0 0]\n", "[ 0 -((a^2 - b^2)*mu^2 + b^2 - 1)*r/(sqrt(-a^2*b^2 - (a^2 - b^2)*mu^2 - ((a^2 - b^2)*mu^2 + b^2 - 1)*r^2 + a^2)*sqrt(a + 1)*sqrt(-a + 1)*sqrt(b + 1)*sqrt(-b + 1)) -((a^2 - b^2)*mu*r^2 + (a^2 - b^2)*mu)/(sqrt(-a^2*b^2 - (a^2 - b^2)*mu^2 - ((a^2 - b^2)*mu^2 + b^2 - 1)*r^2 + a^2)*sqrt(a + 1)*sqrt(-a + 1)*sqrt(b + 1)*sqrt(-b + 1)) 0 0]\n", "[ 0 -sqrt(-a^2*b^2 - (a^2 - b^2)*mu^2 - ((a^2 - b^2)*mu^2 + b^2 - 1)*r^2 + a^2)*((b^4 - (a^2 + 1)*b^2 + a^2)*mu^3 - (b^4 - (a^2 + 1)*b^2 + a^2)*mu)*sqrt(a + 1)*sqrt(-a + 1)*r/((a^4*b^4 - 2*a^4*b^2 + (a^4 - 2*a^2*b^2 + b^4)*mu^4 + ((a^4 - 2*a^2*b^2 + b^4)*mu^4 + b^4 - 2*(b^4 - (a^2 + 1)*b^2 + a^2)*mu^2 - 2*b^2 + 1)*r^4 + a^4 - 2*(a^2*b^4 + a^4 - (a^4 + a^2)*b^2)*mu^2 + 2*(a^2*b^4 + (a^4 - 2*a^2*b^2 + b^4)*mu^4 - 2*a^2*b^2 - ((a^2 + 1)*b^4 + a^4 - (a^4 + 2*a^2 + 1)*b^2 + a^2)*mu^2 + a^2)*r^2)*sqrt(b^2 + r^2)) -(a^2*b^2 + (b^2 - 1)*r^2 - a^2)*sqrt(-a^2*b^2 - (a^2 - b^2)*mu^2 - ((a^2 - b^2)*mu^2 + b^2 - 1)*r^2 + a^2)*sqrt(b^2 + r^2)*sqrt(a + 1)*sqrt(-a + 1)/(a^4*b^4 - 2*a^4*b^2 + (a^4 - 2*a^2*b^2 + b^4)*mu^4 + ((a^4 - 2*a^2*b^2 + b^4)*mu^4 + b^4 - 2*(b^4 - (a^2 + 1)*b^2 + a^2)*mu^2 - 2*b^2 + 1)*r^4 + a^4 - 2*(a^2*b^4 + a^4 - (a^4 + a^2)*b^2)*mu^2 + 2*(a^2*b^4 + (a^4 - 2*a^2*b^2 + b^4)*mu^4 - 2*a^2*b^2 - ((a^2 + 1)*b^4 + a^4 - (a^4 + 2*a^2 + 1)*b^2 + a^2)*mu^2 + a^2)*r^2) 0 0]\n", "[ a 0 0 1 0]\n", "[ b 0 0 0 1]" ] }, "execution_count": 48, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL_to_ADS.jacobian()" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n", "1 & 0 & 0 & 0 & 0 \\\\\n", "0 & \\frac{\\sqrt{2} {\\left({\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{3} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y + {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y\\right)}}{2 \\, \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} \\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} - b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}}} & \\frac{\\sqrt{2} {\\left({\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{3} - {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}\\right)} y^{4} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left(a^{2} - b^{2}\\right)} {\\chi} y^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi} y^{2}\\right)}}{2 \\, \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} \\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} - b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}}} & 0 & 0 \\\\\n", "0 & -\\frac{{\\left(\\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left(\\sqrt{2} a^{2} - \\sqrt{2} b^{2}\\right)} \\sqrt{b + 1} \\sqrt{-b + 1} {\\chi} + {\\left({\\left({\\left(\\sqrt{2} a^{4} - \\sqrt{2} b^{4} - 2 \\, \\sqrt{2} a^{2} + 2 \\, \\sqrt{2} b^{2}\\right)} {\\chi}^{3} - {\\left(\\sqrt{2} a^{4} - {\\left(\\sqrt{2} a^{2} - \\sqrt{2}\\right)} b^{2} - \\sqrt{2} a^{2}\\right)} {\\chi}\\right)} y^{2} - {\\left(\\sqrt{2} a^{4} - 2 \\, \\sqrt{2} a^{2} b^{2} + \\sqrt{2} b^{4}\\right)} {\\chi}\\right)} \\sqrt{b + 1} \\sqrt{-b + 1}\\right)} \\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} + b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}}}{2 \\, {\\left({\\left({\\left(a^{6} - 3 \\, a^{4} b^{2} + 3 \\, a^{2} b^{4} - b^{6}\\right)} {\\chi}^{6} - a^{6} - 3 \\, {\\left(a^{6} + {\\left(a^{2} - 1\\right)} b^{4} - a^{4} - 2 \\, {\\left(a^{4} - a^{2}\\right)} b^{2}\\right)} {\\chi}^{4} + 3 \\, a^{4} + 3 \\, {\\left(a^{6} - 2 \\, a^{4} - {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\chi}^{2} - 3 \\, a^{2} + 1\\right)} y^{6} - a^{6} + 3 \\, a^{4} b^{2} - 3 \\, a^{2} b^{4} + b^{6} - {\\left(3 \\, a^{6} + {\\left(3 \\, a^{6} + b^{6} + {\\left(a^{2} - 4\\right)} b^{4} - 4 \\, a^{4} - {\\left(5 \\, a^{4} - 8 \\, a^{2}\\right)} b^{2}\\right)} {\\chi}^{4} - 6 \\, a^{4} - 3 \\, {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} - 2 \\, {\\left(3 \\, a^{6} + {\\left(a^{2} - 1\\right)} b^{4} - 5 \\, a^{4} - 2 \\, {\\left(2 \\, a^{4} - 3 \\, a^{2} + 1\\right)} b^{2} + 2 \\, a^{2}\\right)} {\\chi}^{2} + 3 \\, a^{2}\\right)} y^{4} - {\\left(3 \\, a^{6} + 3 \\, {\\left(a^{2} - 1\\right)} b^{4} - 3 \\, a^{4} - 6 \\, {\\left(a^{4} - a^{2}\\right)} b^{2} - {\\left(3 \\, a^{6} + b^{6} + {\\left(a^{2} - 4\\right)} b^{4} - 4 \\, a^{4} - {\\left(5 \\, a^{4} - 8 \\, a^{2}\\right)} b^{2}\\right)} {\\chi}^{2}\\right)} y^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left({\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}\\right)}\\right)}} & -\\frac{\\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} + b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}} {\\left(\\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left({\\left(\\sqrt{2} a^{2} - \\sqrt{2}\\right)} y^{3} + {\\left(\\sqrt{2} a^{2} - \\sqrt{2} b^{2}\\right)} y\\right)} \\sqrt{b + 1} \\sqrt{-b + 1} - {\\left({\\left(\\sqrt{2} a^{4} - {\\left(\\sqrt{2} a^{4} - {\\left(\\sqrt{2} a^{2} - \\sqrt{2}\\right)} b^{2} - \\sqrt{2} a^{2}\\right)} {\\chi}^{2} - 2 \\, \\sqrt{2} a^{2} + \\sqrt{2}\\right)} y^{5} + {\\left(2 \\, \\sqrt{2} a^{4} - 2 \\, {\\left(\\sqrt{2} a^{2} - \\sqrt{2}\\right)} b^{2} - {\\left(\\sqrt{2} a^{4} - \\sqrt{2} b^{4} - 2 \\, \\sqrt{2} a^{2} + 2 \\, \\sqrt{2} b^{2}\\right)} {\\chi}^{2} - 2 \\, \\sqrt{2} a^{2}\\right)} y^{3} + {\\left(\\sqrt{2} a^{4} - 2 \\, \\sqrt{2} a^{2} b^{2} + \\sqrt{2} b^{4}\\right)} y\\right)} \\sqrt{b + 1} \\sqrt{-b + 1}\\right)}}{2 \\, {\\left({\\left({\\left(a^{6} - 3 \\, a^{4} b^{2} + 3 \\, a^{2} b^{4} - b^{6}\\right)} {\\chi}^{6} - a^{6} - 3 \\, {\\left(a^{6} + {\\left(a^{2} - 1\\right)} b^{4} - a^{4} - 2 \\, {\\left(a^{4} - a^{2}\\right)} b^{2}\\right)} {\\chi}^{4} + 3 \\, a^{4} + 3 \\, {\\left(a^{6} - 2 \\, a^{4} - {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\chi}^{2} - 3 \\, a^{2} + 1\\right)} y^{6} - a^{6} + 3 \\, a^{4} b^{2} - 3 \\, a^{2} b^{4} + b^{6} - {\\left(3 \\, a^{6} + {\\left(3 \\, a^{6} + b^{6} + {\\left(a^{2} - 4\\right)} b^{4} - 4 \\, a^{4} - {\\left(5 \\, a^{4} - 8 \\, a^{2}\\right)} b^{2}\\right)} {\\chi}^{4} - 6 \\, a^{4} - 3 \\, {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} - 2 \\, {\\left(3 \\, a^{6} + {\\left(a^{2} - 1\\right)} b^{4} - 5 \\, a^{4} - 2 \\, {\\left(2 \\, a^{4} - 3 \\, a^{2} + 1\\right)} b^{2} + 2 \\, a^{2}\\right)} {\\chi}^{2} + 3 \\, a^{2}\\right)} y^{4} - {\\left(3 \\, a^{6} + 3 \\, {\\left(a^{2} - 1\\right)} b^{4} - 3 \\, a^{4} - 6 \\, {\\left(a^{4} - a^{2}\\right)} b^{2} - {\\left(3 \\, a^{6} + b^{6} + {\\left(a^{2} - 4\\right)} b^{4} - 4 \\, a^{4} - {\\left(5 \\, a^{4} - 8 \\, a^{2}\\right)} b^{2}\\right)} {\\chi}^{2}\\right)} y^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left({\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}\\right)}\\right)}} & 0 & 0 \\\\\n", "-a & 0 & 0 & 1 & 0 \\\\\n", "-b & 0 & 0 & 0 & 1\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n", "1 & 0 & 0 & 0 & 0 \\\\\n", "0 & \\frac{\\sqrt{2} {\\left({\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{3} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y + {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y\\right)}}{2 \\, \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} \\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} - b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}}} & \\frac{\\sqrt{2} {\\left({\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{3} - {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}\\right)} y^{4} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left(a^{2} - b^{2}\\right)} {\\chi} y^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi} y^{2}\\right)}}{2 \\, \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} \\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} - b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}}} & 0 & 0 \\\\\n", "0 & -\\frac{{\\left(\\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left(\\sqrt{2} a^{2} - \\sqrt{2} b^{2}\\right)} \\sqrt{b + 1} \\sqrt{-b + 1} {\\chi} + {\\left({\\left({\\left(\\sqrt{2} a^{4} - \\sqrt{2} b^{4} - 2 \\, \\sqrt{2} a^{2} + 2 \\, \\sqrt{2} b^{2}\\right)} {\\chi}^{3} - {\\left(\\sqrt{2} a^{4} - {\\left(\\sqrt{2} a^{2} - \\sqrt{2}\\right)} b^{2} - \\sqrt{2} a^{2}\\right)} {\\chi}\\right)} y^{2} - {\\left(\\sqrt{2} a^{4} - 2 \\, \\sqrt{2} a^{2} b^{2} + \\sqrt{2} b^{4}\\right)} {\\chi}\\right)} \\sqrt{b + 1} \\sqrt{-b + 1}\\right)} \\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} + b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}}}{2 \\, {\\left({\\left({\\left(a^{6} - 3 \\, a^{4} b^{2} + 3 \\, a^{2} b^{4} - b^{6}\\right)} {\\chi}^{6} - a^{6} - 3 \\, {\\left(a^{6} + {\\left(a^{2} - 1\\right)} b^{4} - a^{4} - 2 \\, {\\left(a^{4} - a^{2}\\right)} b^{2}\\right)} {\\chi}^{4} + 3 \\, a^{4} + 3 \\, {\\left(a^{6} - 2 \\, a^{4} - {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\chi}^{2} - 3 \\, a^{2} + 1\\right)} y^{6} - a^{6} + 3 \\, a^{4} b^{2} - 3 \\, a^{2} b^{4} + b^{6} - {\\left(3 \\, a^{6} + {\\left(3 \\, a^{6} + b^{6} + {\\left(a^{2} - 4\\right)} b^{4} - 4 \\, a^{4} - {\\left(5 \\, a^{4} - 8 \\, a^{2}\\right)} b^{2}\\right)} {\\chi}^{4} - 6 \\, a^{4} - 3 \\, {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} - 2 \\, {\\left(3 \\, a^{6} + {\\left(a^{2} - 1\\right)} b^{4} - 5 \\, a^{4} - 2 \\, {\\left(2 \\, a^{4} - 3 \\, a^{2} + 1\\right)} b^{2} + 2 \\, a^{2}\\right)} {\\chi}^{2} + 3 \\, a^{2}\\right)} y^{4} - {\\left(3 \\, a^{6} + 3 \\, {\\left(a^{2} - 1\\right)} b^{4} - 3 \\, a^{4} - 6 \\, {\\left(a^{4} - a^{2}\\right)} b^{2} - {\\left(3 \\, a^{6} + b^{6} + {\\left(a^{2} - 4\\right)} b^{4} - 4 \\, a^{4} - {\\left(5 \\, a^{4} - 8 \\, a^{2}\\right)} b^{2}\\right)} {\\chi}^{2}\\right)} y^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left({\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}\\right)}\\right)}} & -\\frac{\\sqrt{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1\\right)} y^{2} - a^{2} + b^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}}} {\\left(\\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left({\\left(\\sqrt{2} a^{2} - \\sqrt{2}\\right)} y^{3} + {\\left(\\sqrt{2} a^{2} - \\sqrt{2} b^{2}\\right)} y\\right)} \\sqrt{b + 1} \\sqrt{-b + 1} - {\\left({\\left(\\sqrt{2} a^{4} - {\\left(\\sqrt{2} a^{4} - {\\left(\\sqrt{2} a^{2} - \\sqrt{2}\\right)} b^{2} - \\sqrt{2} a^{2}\\right)} {\\chi}^{2} - 2 \\, \\sqrt{2} a^{2} + \\sqrt{2}\\right)} y^{5} + {\\left(2 \\, \\sqrt{2} a^{4} - 2 \\, {\\left(\\sqrt{2} a^{2} - \\sqrt{2}\\right)} b^{2} - {\\left(\\sqrt{2} a^{4} - \\sqrt{2} b^{4} - 2 \\, \\sqrt{2} a^{2} + 2 \\, \\sqrt{2} b^{2}\\right)} {\\chi}^{2} - 2 \\, \\sqrt{2} a^{2}\\right)} y^{3} + {\\left(\\sqrt{2} a^{4} - 2 \\, \\sqrt{2} a^{2} b^{2} + \\sqrt{2} b^{4}\\right)} y\\right)} \\sqrt{b + 1} \\sqrt{-b + 1}\\right)}}{2 \\, {\\left({\\left({\\left(a^{6} - 3 \\, a^{4} b^{2} + 3 \\, a^{2} b^{4} - b^{6}\\right)} {\\chi}^{6} - a^{6} - 3 \\, {\\left(a^{6} + {\\left(a^{2} - 1\\right)} b^{4} - a^{4} - 2 \\, {\\left(a^{4} - a^{2}\\right)} b^{2}\\right)} {\\chi}^{4} + 3 \\, a^{4} + 3 \\, {\\left(a^{6} - 2 \\, a^{4} - {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} + a^{2}\\right)} {\\chi}^{2} - 3 \\, a^{2} + 1\\right)} y^{6} - a^{6} + 3 \\, a^{4} b^{2} - 3 \\, a^{2} b^{4} + b^{6} - {\\left(3 \\, a^{6} + {\\left(3 \\, a^{6} + b^{6} + {\\left(a^{2} - 4\\right)} b^{4} - 4 \\, a^{4} - {\\left(5 \\, a^{4} - 8 \\, a^{2}\\right)} b^{2}\\right)} {\\chi}^{4} - 6 \\, a^{4} - 3 \\, {\\left(a^{4} - 2 \\, a^{2} + 1\\right)} b^{2} - 2 \\, {\\left(3 \\, a^{6} + {\\left(a^{2} - 1\\right)} b^{4} - 5 \\, a^{4} - 2 \\, {\\left(2 \\, a^{4} - 3 \\, a^{2} + 1\\right)} b^{2} + 2 \\, a^{2}\\right)} {\\chi}^{2} + 3 \\, a^{2}\\right)} y^{4} - {\\left(3 \\, a^{6} + 3 \\, {\\left(a^{2} - 1\\right)} b^{4} - 3 \\, a^{4} - 6 \\, {\\left(a^{4} - a^{2}\\right)} b^{2} - {\\left(3 \\, a^{6} + b^{6} + {\\left(a^{2} - 4\\right)} b^{4} - 4 \\, a^{4} - {\\left(5 \\, a^{4} - 8 \\, a^{2}\\right)} b^{2}\\right)} {\\chi}^{2}\\right)} y^{2} + \\sqrt{{\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - b^{4} - 2 \\, a^{2} + 2 \\, b^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}} {\\left({\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 2 \\, a^{2} + 1\\right)} y^{4} + a^{4} - 2 \\, a^{2} b^{2} + b^{4} + 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - a^{2}\\right)} y^{2}\\right)}\\right)}} & 0 & 0 \\\\\n", "-a & 0 & 0 & 1 & 0 \\\\\n", "-b & 0 & 0 & 0 & 1\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ 1 0 0 0 0]\n", "[ 0 1/2*sqrt(2)*(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^3 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2)*((a^2 - b^2)*ch^2 - a^2 + 1)*y + (a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y)/(sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2)*sqrt(((a^2 - b^2)*ch^2 - a^2 + 1)*y^2 - a^2 - b^2 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2))) 1/2*sqrt(2)*(((a^4 - 2*a^2*b^2 + b^4)*ch^3 - (a^4 - (a^2 - 1)*b^2 - a^2)*ch)*y^4 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2)*(a^2 - b^2)*ch*y^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch*y^2)/(sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2)*sqrt(((a^2 - b^2)*ch^2 - a^2 + 1)*y^2 - a^2 - b^2 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2))) 0 0]\n", "[ 0 -1/2*(sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2)*(sqrt(2)*a^2 - sqrt(2)*b^2)*sqrt(b + 1)*sqrt(-b + 1)*ch + (((sqrt(2)*a^4 - sqrt(2)*b^4 - 2*sqrt(2)*a^2 + 2*sqrt(2)*b^2)*ch^3 - (sqrt(2)*a^4 - (sqrt(2)*a^2 - sqrt(2))*b^2 - sqrt(2)*a^2)*ch)*y^2 - (sqrt(2)*a^4 - 2*sqrt(2)*a^2*b^2 + sqrt(2)*b^4)*ch)*sqrt(b + 1)*sqrt(-b + 1))*sqrt(((a^2 - b^2)*ch^2 - a^2 + 1)*y^2 - a^2 + b^2 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2))/(((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*ch^6 - a^6 - 3*(a^6 + (a^2 - 1)*b^4 - a^4 - 2*(a^4 - a^2)*b^2)*ch^4 + 3*a^4 + 3*(a^6 - 2*a^4 - (a^4 - 2*a^2 + 1)*b^2 + a^2)*ch^2 - 3*a^2 + 1)*y^6 - a^6 + 3*a^4*b^2 - 3*a^2*b^4 + b^6 - (3*a^6 + (3*a^6 + b^6 + (a^2 - 4)*b^4 - 4*a^4 - (5*a^4 - 8*a^2)*b^2)*ch^4 - 6*a^4 - 3*(a^4 - 2*a^2 + 1)*b^2 - 2*(3*a^6 + (a^2 - 1)*b^4 - 5*a^4 - 2*(2*a^4 - 3*a^2 + 1)*b^2 + 2*a^2)*ch^2 + 3*a^2)*y^4 - (3*a^6 + 3*(a^2 - 1)*b^4 - 3*a^4 - 6*(a^4 - a^2)*b^2 - (3*a^6 + b^6 + (a^2 - 4)*b^4 - 4*a^4 - (5*a^4 - 8*a^2)*b^2)*ch^2)*y^2 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2)*(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - a^2)*y^2)) -1/2*sqrt(((a^2 - b^2)*ch^2 - a^2 + 1)*y^2 - a^2 + b^2 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2))*(sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2)*((sqrt(2)*a^2 - sqrt(2))*y^3 + (sqrt(2)*a^2 - sqrt(2)*b^2)*y)*sqrt(b + 1)*sqrt(-b + 1) - ((sqrt(2)*a^4 - (sqrt(2)*a^4 - (sqrt(2)*a^2 - sqrt(2))*b^2 - sqrt(2)*a^2)*ch^2 - 2*sqrt(2)*a^2 + sqrt(2))*y^5 + (2*sqrt(2)*a^4 - 2*(sqrt(2)*a^2 - sqrt(2))*b^2 - (sqrt(2)*a^4 - sqrt(2)*b^4 - 2*sqrt(2)*a^2 + 2*sqrt(2)*b^2)*ch^2 - 2*sqrt(2)*a^2)*y^3 + (sqrt(2)*a^4 - 2*sqrt(2)*a^2*b^2 + sqrt(2)*b^4)*y)*sqrt(b + 1)*sqrt(-b + 1))/(((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*ch^6 - a^6 - 3*(a^6 + (a^2 - 1)*b^4 - a^4 - 2*(a^4 - a^2)*b^2)*ch^4 + 3*a^4 + 3*(a^6 - 2*a^4 - (a^4 - 2*a^2 + 1)*b^2 + a^2)*ch^2 - 3*a^2 + 1)*y^6 - a^6 + 3*a^4*b^2 - 3*a^2*b^4 + b^6 - (3*a^6 + (3*a^6 + b^6 + (a^2 - 4)*b^4 - 4*a^4 - (5*a^4 - 8*a^2)*b^2)*ch^4 - 6*a^4 - 3*(a^4 - 2*a^2 + 1)*b^2 - 2*(3*a^6 + (a^2 - 1)*b^4 - 5*a^4 - 2*(2*a^4 - 3*a^2 + 1)*b^2 + 2*a^2)*ch^2 + 3*a^2)*y^4 - (3*a^6 + 3*(a^2 - 1)*b^4 - 3*a^4 - 6*(a^4 - a^2)*b^2 - (3*a^6 + b^6 + (a^2 - 4)*b^4 - 4*a^4 - (5*a^4 - 8*a^2)*b^2)*ch^2)*y^2 + sqrt(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - b^4 - 2*a^2 + 2*b^2)*ch^2 - a^2)*y^2)*(((a^4 - 2*a^2*b^2 + b^4)*ch^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - 2*a^2 + 1)*y^4 + a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - (a^2 - 1)*b^2 - (a^4 - (a^2 - 1)*b^2 - a^2)*ch^2 - a^2)*y^2)) 0 0]\n", "[ -a 0 0 1 0]\n", "[ -b 0 0 0 1]" ] }, "execution_count": 49, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL_to_ADS.inverse().jacobian()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "*Remark:* despite the rather complicated relation between $y$ and $(r,\\mu)$, the ratio $(1 + r^2)/(1 + y^2)$ depends only on $\\mu$ and takes a simple form:\n", "$$\n", " \\frac{1 + r^2}{1 + y^2} = \\frac{(1 - a^2) (1 - b^2)}{1 - a^2\\mu^2 - b^2 (1 - \\mu^2)}\n", "$$\n", "Indeed:" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(a + 1\\right)} {\\left(a - 1\\right)} {\\left(b + 1\\right)} {\\left(b - 1\\right)}}{a^{2} {\\mu}^{2} - b^{2} {\\mu}^{2} + b^{2} - 1}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(a + 1\\right)} {\\left(a - 1\\right)} {\\left(b + 1\\right)} {\\left(b - 1\\right)}}{a^{2} {\\mu}^{2} - b^{2} {\\mu}^{2} + b^{2} - 1}$$" ], "text/plain": [ "-(a + 1)*(a - 1)*(b + 1)*(b - 1)/(a^2*mu^2 - b^2*mu^2 + b^2 - 1)" ] }, "execution_count": 50, "metadata": { }, "output_type": "execute_result" } ], "source": [ "((1 + r^2)/(1 + ys^2)).simplify_full().factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Metric components in global ADS coordinates" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "For generic values of $(a,b)$, Sage does not succeed in computing the components of the \n", "metric tensor $G$ in a reasonable time.\n", "Only for $a=b$ or $b=0$ it manages to do so. For $b=0$, the expression is cumbersome, but\n", "for $a=b$, one gets a rather simple expression:" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "if a == b:\n", " show(G.display_comp(chart=ADS, only_nonredundant=True))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Asymptotic form of the metric in ADS coordinates (check of Eq. (2.17))\n", "\n", "For $y\\to +\\infty$, the metric tensor $G$ can be approximated by $K$, the expression of the latter in ADS coordinates being given by Eq. (3.27) of [Gibbons, Perry & Pope, CQG **22**, 1503 (2005)](https://doi.org/10.1088/0264-9381/22/9/002) ([arXiv:hep-th/0408217](https://arxiv.org/abs/hep-th/0408217)) (our Eq. (2.17)):" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "M.set_default_frame(ADS.frame())\n", "M.set_default_chart(ADS)" ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}K = \\left( -y^{2} + \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} - 1 \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\mathrm{d} T\\otimes \\mathrm{d} {\\Phi} -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} T\\otimes \\mathrm{d} {\\Psi} + \\left( \\frac{1}{y^{2} - \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( -\\frac{y^{2}}{{\\chi}^{2} - 1} \\right) \\mathrm{d} {\\chi}\\otimes \\mathrm{d} {\\chi} + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} T + \\left( -{\\left({\\chi}^{2} - 1\\right)} y^{2} + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)}^{2} a^{2} m}{{\\Delta}^{3} y^{2}} \\right) \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} {\\Phi} -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} {\\Psi} -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} T -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Phi} + \\left( {\\chi}^{2} y^{2} + \\frac{2 \\, b^{2} {\\chi}^{4} m}{{\\Delta}^{3} y^{2}} \\right) \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K = \\left( -y^{2} + \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} - 1 \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\mathrm{d} T\\otimes \\mathrm{d} {\\Phi} -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} T\\otimes \\mathrm{d} {\\Psi} + \\left( \\frac{1}{y^{2} - \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( -\\frac{y^{2}}{{\\chi}^{2} - 1} \\right) \\mathrm{d} {\\chi}\\otimes \\mathrm{d} {\\chi} + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} T + \\left( -{\\left({\\chi}^{2} - 1\\right)} y^{2} + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)}^{2} a^{2} m}{{\\Delta}^{3} y^{2}} \\right) \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} {\\Phi} -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} {\\Psi} -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} T -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Phi} + \\left( {\\chi}^{2} y^{2} + \\frac{2 \\, b^{2} {\\chi}^{4} m}{{\\Delta}^{3} y^{2}} \\right) \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Psi}$$" ], "text/plain": [ "K = (-y^2 + 2*m/(Delta^3*y^2) - 1) dT*dT + 2*(ch^2 - 1)*a*m/(Delta^3*y^2) dT*dPh - 2*b*ch^2*m/(Delta^3*y^2) dT*dPs + 1/(y^2 - 2*m/(Delta^3*y^2) + 1) dy*dy - y^2/(ch^2 - 1) dch*dch + 2*(ch^2 - 1)*a*m/(Delta^3*y^2) dPh*dT + (-(ch^2 - 1)*y^2 + 2*(ch^2 - 1)^2*a^2*m/(Delta^3*y^2)) dPh*dPh - 2*(ch^2 - 1)*a*b*ch^2*m/(Delta^3*y^2) dPh*dPs - 2*b*ch^2*m/(Delta^3*y^2) dPs*dT - 2*(ch^2 - 1)*a*b*ch^2*m/(Delta^3*y^2) dPs*dPh + (ch^2*y^2 + 2*b^2*ch^4*m/(Delta^3*y^2)) dPs*dPs" ] }, "execution_count": 53, "metadata": { }, "output_type": "execute_result" } ], "source": [ "K = M.lorentzian_metric('K')\n", "Delta = var('Delta', latex_name=r'\\Delta', domain='real')\n", "K[0, 0] = -1 - y^2 + 2*m/(Delta^3*y^2)\n", "K[0, 3] = -2*m*a*(1 - ch^2)/(Delta^3*y^2)\n", "K[0, 4] = -2*m*b*ch^2/(Delta^3*y^2)\n", "K[1, 1] = 1/(1 + y^2 - 2*m/(Delta^3*y^2))\n", "K[2, 2] = y^2/(1 - ch^2)\n", "K[3, 3] = y^2*(1 - ch^2) + 2*m*a^2*(1 - ch^2)^2/(Delta^3*y^2)\n", "K[3, 4] = 2*m*a*b*ch^2*(1 - ch^2)/(Delta^3*y^2)\n", "K[4, 4] = y^2*ch^2 + 2*m*b^2*ch^4/(Delta^3*y^2)\n", "K.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In the above expression, we do not have specified $\\Delta$. Its explicit expression in terms of $a$, $b$ and $\\chi$ is" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(a^{2} - b^{2}\\right)} {\\chi}^{2} - a^{2} + 1$$" ], "text/plain": [ "(a^2 - b^2)*ch^2 - a^2 + 1" ] }, "execution_count": 54, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Delta_abc = (1 - a^2*(1 - ch^2) - b^2*ch^2).simplify_full()\n", "Delta_abc" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (2.17) for $a=b$:" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "For $a=b$, $G$ and $K$ differ only by a term proportional to $\\mathrm{d}y\\otimes\\mathrm{d}y$:" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "if a == b:\n", " K0 = K.copy(name='K')\n", " K0.apply_map(lambda x: x.subs({Delta: Delta_abc}))\n", " GmK = G - K0\n", " show(GmK.display())" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This difference is only of order $O(1/y^6)$:" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "if a == b:\n", " es = sum(asympt(GmK[1,1].expr(), y, k) for k in range(9))\n", " show(es)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Setting the metric to its asymptotic form:" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -y^{2} + \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} - 1 \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\mathrm{d} T\\otimes \\mathrm{d} {\\Phi} -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} T\\otimes \\mathrm{d} {\\Psi} + \\left( \\frac{1}{y^{2} - \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( -\\frac{y^{2}}{{\\chi}^{2} - 1} \\right) \\mathrm{d} {\\chi}\\otimes \\mathrm{d} {\\chi} + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} T + \\left( -{\\left({\\chi}^{2} - 1\\right)} y^{2} + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)}^{2} a^{2} m}{{\\Delta}^{3} y^{2}} \\right) \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} {\\Phi} -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} {\\Psi} -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} T -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Phi} + \\left( {\\chi}^{2} y^{2} + \\frac{2 \\, b^{2} {\\chi}^{4} m}{{\\Delta}^{3} y^{2}} \\right) \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -y^{2} + \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} - 1 \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\mathrm{d} T\\otimes \\mathrm{d} {\\Phi} -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} T\\otimes \\mathrm{d} {\\Psi} + \\left( \\frac{1}{y^{2} - \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( -\\frac{y^{2}}{{\\chi}^{2} - 1} \\right) \\mathrm{d} {\\chi}\\otimes \\mathrm{d} {\\chi} + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} T + \\left( -{\\left({\\chi}^{2} - 1\\right)} y^{2} + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)}^{2} a^{2} m}{{\\Delta}^{3} y^{2}} \\right) \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} {\\Phi} -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Phi}\\otimes \\mathrm{d} {\\Psi} -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} T -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Phi} + \\left( {\\chi}^{2} y^{2} + \\frac{2 \\, b^{2} {\\chi}^{4} m}{{\\Delta}^{3} y^{2}} \\right) \\mathrm{d} {\\Psi}\\otimes \\mathrm{d} {\\Psi}$$" ], "text/plain": [ "G = (-y^2 + 2*m/(Delta^3*y^2) - 1) dT*dT + 2*(ch^2 - 1)*a*m/(Delta^3*y^2) dT*dPh - 2*b*ch^2*m/(Delta^3*y^2) dT*dPs + 1/(y^2 - 2*m/(Delta^3*y^2) + 1) dy*dy - y^2/(ch^2 - 1) dch*dch + 2*(ch^2 - 1)*a*m/(Delta^3*y^2) dPh*dT + (-(ch^2 - 1)*y^2 + 2*(ch^2 - 1)^2*a^2*m/(Delta^3*y^2)) dPh*dPh - 2*(ch^2 - 1)*a*b*ch^2*m/(Delta^3*y^2) dPh*dPs - 2*b*ch^2*m/(Delta^3*y^2) dPs*dT - 2*(ch^2 - 1)*a*b*ch^2*m/(Delta^3*y^2) dPs*dPh + (ch^2*y^2 + 2*b^2*ch^4*m/(Delta^3*y^2)) dPs*dPs" ] }, "execution_count": 57, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.set(K)\n", "G.display()" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, T \\, T }^{ \\phantom{\\, T}\\phantom{\\, T} } & = & -y^{2} + \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} - 1 \\\\ G_{ \\, T \\, {\\Phi} }^{ \\phantom{\\, T}\\phantom{\\, {\\Phi}} } & = & \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, T \\, {\\Psi} }^{ \\phantom{\\, T}\\phantom{\\, {\\Psi}} } & = & -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{1}{y^{2} - \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} + 1} \\\\ G_{ \\, {\\chi} \\, {\\chi} }^{ \\phantom{\\, {\\chi}}\\phantom{\\, {\\chi}} } & = & -\\frac{y^{2}}{{\\chi}^{2} - 1} \\\\ G_{ \\, {\\Phi} \\, T }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, T} } & = & \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Phi} \\, {\\Phi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Phi}} } & = & -{\\left({\\chi}^{2} - 1\\right)} y^{2} + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)}^{2} a^{2} m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Phi} \\, {\\Psi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Psi}} } & = & -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Psi} \\, T }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, T} } & = & -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Phi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & {\\chi}^{2} y^{2} + \\frac{2 \\, b^{2} {\\chi}^{4} m}{{\\Delta}^{3} y^{2}} \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, T \\, T }^{ \\phantom{\\, T}\\phantom{\\, T} } & = & -y^{2} + \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} - 1 \\\\ G_{ \\, T \\, {\\Phi} }^{ \\phantom{\\, T}\\phantom{\\, {\\Phi}} } & = & \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, T \\, {\\Psi} }^{ \\phantom{\\, T}\\phantom{\\, {\\Psi}} } & = & -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{1}{y^{2} - \\frac{2 \\, m}{{\\Delta}^{3} y^{2}} + 1} \\\\ G_{ \\, {\\chi} \\, {\\chi} }^{ \\phantom{\\, {\\chi}}\\phantom{\\, {\\chi}} } & = & -\\frac{y^{2}}{{\\chi}^{2} - 1} \\\\ G_{ \\, {\\Phi} \\, T }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, T} } & = & \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Phi} \\, {\\Phi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Phi}} } & = & -{\\left({\\chi}^{2} - 1\\right)} y^{2} + \\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)}^{2} a^{2} m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Phi} \\, {\\Psi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Psi}} } & = & -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Psi} \\, T }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, T} } & = & -\\frac{2 \\, b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Phi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, {\\left({\\chi}^{2} - 1\\right)} a b {\\chi}^{2} m}{{\\Delta}^{3} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & {\\chi}^{2} y^{2} + \\frac{2 \\, b^{2} {\\chi}^{4} m}{{\\Delta}^{3} y^{2}} \\end{array}$$" ], "text/plain": [ "G_T,T = -y^2 + 2*m/(Delta^3*y^2) - 1 \n", "G_T,Ph = 2*(ch^2 - 1)*a*m/(Delta^3*y^2) \n", "G_T,Ps = -2*b*ch^2*m/(Delta^3*y^2) \n", "G_y,y = 1/(y^2 - 2*m/(Delta^3*y^2) + 1) \n", "G_ch,ch = -y^2/(ch^2 - 1) \n", "G_Ph,T = 2*(ch^2 - 1)*a*m/(Delta^3*y^2) \n", "G_Ph,Ph = -(ch^2 - 1)*y^2 + 2*(ch^2 - 1)^2*a^2*m/(Delta^3*y^2) \n", "G_Ph,Ps = -2*(ch^2 - 1)*a*b*ch^2*m/(Delta^3*y^2) \n", "G_Ps,T = -2*b*ch^2*m/(Delta^3*y^2) \n", "G_Ps,Ph = -2*(ch^2 - 1)*a*b*ch^2*m/(Delta^3*y^2) \n", "G_Ps,Ps = ch^2*y^2 + 2*b^2*ch^4*m/(Delta^3*y^2) " ] }, "execution_count": 58, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display_comp()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## String worldsheet" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The string worldsheet as a 2-dimensional Lorentzian submanifold of $\\mathcal{M}$:" ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-dimensional Lorentzian submanifold W immersed in the 5-dimensional Lorentzian manifold M\n" ] } ], "source": [ "W = Manifold(2, 'W', ambient=M, structure='Lorentzian')\n", "print(W)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Let us assume that the string worldsheet is parametrized by $(T,y)$:" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(W,(T, y)\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(W,(T, y)\\right)$$" ], "text/plain": [ "Chart (W, (T, y))" ] }, "execution_count": 60, "metadata": { }, "output_type": "execute_result" } ], "source": [ "XW. = W.chart(r'T y:(0,+oo)')\n", "XW" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The string embedding in Kerr-AdS spacetime, as an expansion about a \n", "straight string solution in AdS (Eq. (4.27) of the paper):" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} F:& W & \\longrightarrow & \\mathcal{M} \\\\ & \\left(T, y\\right) & \\longmapsto & \\left(T, y, {\\chi}, {\\Phi}, {\\Psi}\\right) = \\left(T, y, {\\left(a + b\\right)}^{2} \\chi_{1}\\left(y\\right) + {\\chi_0}, T a + a \\Phi_{1}\\left(y\\right) + {\\Phi_0}, T b + b \\Psi_{1}\\left(y\\right) + {\\Psi_0}\\right) \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} F:& W & \\longrightarrow & \\mathcal{M} \\\\ & \\left(T, y\\right) & \\longmapsto & \\left(T, y, {\\chi}, {\\Phi}, {\\Psi}\\right) = \\left(T, y, {\\left(a + b\\right)}^{2} \\chi_{1}\\left(y\\right) + {\\chi_0}, T a + a \\Phi_{1}\\left(y\\right) + {\\Phi_0}, T b + b \\Psi_{1}\\left(y\\right) + {\\Psi_0}\\right) \\end{array}$$" ], "text/plain": [ "F: W --> M\n", " (T, y) |--> (T, y, ch, Ph, Ps) = (T, y, (a + b)^2*chi_1(y) + ch0, T*a + a*Phi_1(y) + Phi0, T*b + b*Psi_1(y) + Psi0)" ] }, "execution_count": 61, "metadata": { }, "output_type": "execute_result" } ], "source": [ "ch0 = var('ch0', latex_name=r'\\chi_0', domain='real')\n", "Phi0 = var('Phi0', latex_name=r'\\Phi_0', domain='real')\n", "Psi0 = var('Psi0', latex_name=r'\\Psi_0', domain='real')\n", "\n", "cosTh0 = ch0\n", "sinTh0 = sqrt(1 - ch0^2)\n", "\n", "ch_s = ch0 + (a+b)^2*function('chi_1')(y)\n", "Ph_s = Phi0 + a*T + a*function('Phi_1')(y)\n", "Ps_s = Psi0 + b*T + b*function('Psi_1')(y)\n", "\n", "F = W.diff_map(M, {(XW, ADS): [T, y, ch_s, Ph_s, Ps_s]}, name='F') \n", "\n", "W.set_embedding(F)\n", "\n", "F.display(XW, ADS)" ] }, { "cell_type": "code", "execution_count": 62, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "1 & 0 \\\\\n", "0 & 1 \\\\\n", "0 & {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right) \\\\\n", "a & a \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) \\\\\n", "b & b \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "1 & 0 \\\\\n", "0 & 1 \\\\\n", "0 & {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right) \\\\\n", "a & a \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) \\\\\n", "b & b \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ 1 0]\n", "[ 0 1]\n", "[ 0 (a^2 + 2*a*b + b^2)*diff(chi_1(y), y)]\n", "[ a a*diff(Phi_1(y), y)]\n", "[ b b*diff(Psi_1(y), y)]" ] }, "execution_count": 62, "metadata": { }, "output_type": "execute_result" } ], "source": [ "F.jacobian_matrix()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "### Induced metric on the string worldsheet\n", "\n", "The string worldsheet metric is the metric $g$ induced by the spacetime metric $G$, i.e. the pullback of $G$ by the embedding $F$:" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Because of the bug [#27492](https://trac.sagemath.org/ticket/27492), which impedes parallel computations involving symbolic functions, such as $\\chi_1$, we set back to serial computations:" ] }, { "cell_type": "code", "execution_count": 63, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "Parallelism().set(nproc=1) " ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "g = W.induced_metric()" ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{8 \\, {\\left(a^{10} + 6 \\, a^{9} b + 13 \\, a^{8} b^{2} + 8 \\, a^{7} b^{3} - 14 \\, a^{6} b^{4} - 28 \\, a^{5} b^{5} - 14 \\, a^{4} b^{6} + 8 \\, a^{3} b^{7} + 13 \\, a^{2} b^{8} + 6 \\, a b^{9} + b^{10}\\right)} {\\chi_0} m \\chi_{1}\\left(y\\right)^{3} + 2 \\, {\\left(a^{12} + 8 \\, a^{11} b + 26 \\, a^{10} b^{2} + 40 \\, a^{9} b^{3} + 15 \\, a^{8} b^{4} - 48 \\, a^{7} b^{5} - 84 \\, a^{6} b^{6} - 48 \\, a^{5} b^{7} + 15 \\, a^{4} b^{8} + 40 \\, a^{3} b^{9} + 26 \\, a^{2} b^{10} + 8 \\, a b^{11} + b^{12}\\right)} m \\chi_{1}\\left(y\\right)^{4} - {\\Delta}^{3} y^{2} + {\\left({\\Delta}^{3} a^{2} - {\\Delta}^{3} - {\\left({\\Delta}^{3} a^{2} - {\\Delta}^{3} b^{2}\\right)} {\\chi_0}^{2}\\right)} y^{4} - {\\left({\\left({\\Delta}^{3} a^{6} + 4 \\, {\\Delta}^{3} a^{5} b + 5 \\, {\\Delta}^{3} a^{4} b^{2} - 5 \\, {\\Delta}^{3} a^{2} b^{4} - 4 \\, {\\Delta}^{3} a b^{5} - {\\Delta}^{3} b^{6}\\right)} y^{4} + 4 \\, {\\left(a^{8} - {\\left(a^{2} - 1\\right)} b^{6} - a^{6} - 4 \\, {\\left(a^{3} - a\\right)} b^{5} - 5 \\, {\\left(a^{4} - a^{2}\\right)} b^{4} + 5 \\, {\\left(a^{6} - a^{4}\\right)} b^{2} - 3 \\, {\\left(a^{8} + 4 \\, a^{7} b + 4 \\, a^{6} b^{2} - 4 \\, a^{5} b^{3} - 10 \\, a^{4} b^{4} - 4 \\, a^{3} b^{5} + 4 \\, a^{2} b^{6} + 4 \\, a b^{7} + b^{8}\\right)} {\\chi_0}^{2} + 4 \\, {\\left(a^{7} - a^{5}\\right)} b\\right)} m\\right)} \\chi_{1}\\left(y\\right)^{2} + 2 \\, {\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi_0}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi_0}^{2} - 2 \\, a^{2} + 1\\right)} m - 2 \\, {\\left({\\left({\\Delta}^{3} a^{4} + 2 \\, {\\Delta}^{3} a^{3} b - 2 \\, {\\Delta}^{3} a b^{3} - {\\Delta}^{3} b^{4}\\right)} {\\chi_0} y^{4} - 4 \\, {\\left({\\left(a^{6} + 2 \\, a^{5} b - a^{4} b^{2} - 4 \\, a^{3} b^{3} - a^{2} b^{4} + 2 \\, a b^{5} + b^{6}\\right)} {\\chi_0}^{3} - {\\left(a^{6} - {\\left(a^{2} - 1\\right)} b^{4} - a^{4} - 2 \\, {\\left(a^{3} - a\\right)} b^{3} + 2 \\, {\\left(a^{5} - a^{3}\\right)} b\\right)} {\\chi_0}\\right)} m\\right)} \\chi_{1}\\left(y\\right)}{{\\Delta}^{3} y^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{8 \\, {\\left(a^{10} + 6 \\, a^{9} b + 13 \\, a^{8} b^{2} + 8 \\, a^{7} b^{3} - 14 \\, a^{6} b^{4} - 28 \\, a^{5} b^{5} - 14 \\, a^{4} b^{6} + 8 \\, a^{3} b^{7} + 13 \\, a^{2} b^{8} + 6 \\, a b^{9} + b^{10}\\right)} {\\chi_0} m \\chi_{1}\\left(y\\right)^{3} + 2 \\, {\\left(a^{12} + 8 \\, a^{11} b + 26 \\, a^{10} b^{2} + 40 \\, a^{9} b^{3} + 15 \\, a^{8} b^{4} - 48 \\, a^{7} b^{5} - 84 \\, a^{6} b^{6} - 48 \\, a^{5} b^{7} + 15 \\, a^{4} b^{8} + 40 \\, a^{3} b^{9} + 26 \\, a^{2} b^{10} + 8 \\, a b^{11} + b^{12}\\right)} m \\chi_{1}\\left(y\\right)^{4} - {\\Delta}^{3} y^{2} + {\\left({\\Delta}^{3} a^{2} - {\\Delta}^{3} - {\\left({\\Delta}^{3} a^{2} - {\\Delta}^{3} b^{2}\\right)} {\\chi_0}^{2}\\right)} y^{4} - {\\left({\\left({\\Delta}^{3} a^{6} + 4 \\, {\\Delta}^{3} a^{5} b + 5 \\, {\\Delta}^{3} a^{4} b^{2} - 5 \\, {\\Delta}^{3} a^{2} b^{4} - 4 \\, {\\Delta}^{3} a b^{5} - {\\Delta}^{3} b^{6}\\right)} y^{4} + 4 \\, {\\left(a^{8} - {\\left(a^{2} - 1\\right)} b^{6} - a^{6} - 4 \\, {\\left(a^{3} - a\\right)} b^{5} - 5 \\, {\\left(a^{4} - a^{2}\\right)} b^{4} + 5 \\, {\\left(a^{6} - a^{4}\\right)} b^{2} - 3 \\, {\\left(a^{8} + 4 \\, a^{7} b + 4 \\, a^{6} b^{2} - 4 \\, a^{5} b^{3} - 10 \\, a^{4} b^{4} - 4 \\, a^{3} b^{5} + 4 \\, a^{2} b^{6} + 4 \\, a b^{7} + b^{8}\\right)} {\\chi_0}^{2} + 4 \\, {\\left(a^{7} - a^{5}\\right)} b\\right)} m\\right)} \\chi_{1}\\left(y\\right)^{2} + 2 \\, {\\left({\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi_0}^{4} + a^{4} - 2 \\, {\\left(a^{4} - {\\left(a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi_0}^{2} - 2 \\, a^{2} + 1\\right)} m - 2 \\, {\\left({\\left({\\Delta}^{3} a^{4} + 2 \\, {\\Delta}^{3} a^{3} b - 2 \\, {\\Delta}^{3} a b^{3} - {\\Delta}^{3} b^{4}\\right)} {\\chi_0} y^{4} - 4 \\, {\\left({\\left(a^{6} + 2 \\, a^{5} b - a^{4} b^{2} - 4 \\, a^{3} b^{3} - a^{2} b^{4} + 2 \\, a b^{5} + b^{6}\\right)} {\\chi_0}^{3} - {\\left(a^{6} - {\\left(a^{2} - 1\\right)} b^{4} - a^{4} - 2 \\, {\\left(a^{3} - a\\right)} b^{3} + 2 \\, {\\left(a^{5} - a^{3}\\right)} b\\right)} {\\chi_0}\\right)} m\\right)} \\chi_{1}\\left(y\\right)}{{\\Delta}^{3} y^{2}}$$" ], "text/plain": [ "(8*(a^10 + 6*a^9*b + 13*a^8*b^2 + 8*a^7*b^3 - 14*a^6*b^4 - 28*a^5*b^5 - 14*a^4*b^6 + 8*a^3*b^7 + 13*a^2*b^8 + 6*a*b^9 + b^10)*ch0*m*chi_1(y)^3 + 2*(a^12 + 8*a^11*b + 26*a^10*b^2 + 40*a^9*b^3 + 15*a^8*b^4 - 48*a^7*b^5 - 84*a^6*b^6 - 48*a^5*b^7 + 15*a^4*b^8 + 40*a^3*b^9 + 26*a^2*b^10 + 8*a*b^11 + b^12)*m*chi_1(y)^4 - Delta^3*y^2 + (Delta^3*a^2 - Delta^3 - (Delta^3*a^2 - Delta^3*b^2)*ch0^2)*y^4 - ((Delta^3*a^6 + 4*Delta^3*a^5*b + 5*Delta^3*a^4*b^2 - 5*Delta^3*a^2*b^4 - 4*Delta^3*a*b^5 - Delta^3*b^6)*y^4 + 4*(a^8 - (a^2 - 1)*b^6 - a^6 - 4*(a^3 - a)*b^5 - 5*(a^4 - a^2)*b^4 + 5*(a^6 - a^4)*b^2 - 3*(a^8 + 4*a^7*b + 4*a^6*b^2 - 4*a^5*b^3 - 10*a^4*b^4 - 4*a^3*b^5 + 4*a^2*b^6 + 4*a*b^7 + b^8)*ch0^2 + 4*(a^7 - a^5)*b)*m)*chi_1(y)^2 + 2*((a^4 - 2*a^2*b^2 + b^4)*ch0^4 + a^4 - 2*(a^4 - (a^2 - 1)*b^2 - a^2)*ch0^2 - 2*a^2 + 1)*m - 2*((Delta^3*a^4 + 2*Delta^3*a^3*b - 2*Delta^3*a*b^3 - Delta^3*b^4)*ch0*y^4 - 4*((a^6 + 2*a^5*b - a^4*b^2 - 4*a^3*b^3 - a^2*b^4 + 2*a*b^5 + b^6)*ch0^3 - (a^6 - (a^2 - 1)*b^4 - a^4 - 2*(a^3 - a)*b^3 + 2*(a^5 - a^3)*b)*ch0)*m)*chi_1(y))/(Delta^3*y^2)" ] }, "execution_count": 65, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g[0,0]" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Nambu-Goto action" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The determinant of $g$ is" ] }, { "cell_type": "code", "execution_count": 66, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg = g.determinant().expr()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us define a function for expansions in $a$ and $b$ up to a given order:" ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "eps = var('eps', latex_name=r'\\epsilon', domain='real')\n", "def expand_ab(expr, order):\n", " res = expr.subs({a: eps*a, b: eps*b})\n", " res = res.series(eps, order+1).truncate()\n", " res = res.subs({eps: 1}).simplify_full()\n", " return res" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + 3 \\, a^{4} - 3 \\, {\\left(2 \\, a^{4} - {\\left(2 \\, a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 3 \\, a^{2} + 1$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + 3 \\, a^{4} - 3 \\, {\\left(2 \\, a^{4} - {\\left(2 \\, a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 3 \\, a^{2} + 1$$" ], "text/plain": [ "3*(a^4 - 2*a^2*b^2 + b^4)*ch^4 + 3*a^4 - 3*(2*a^4 - (2*a^2 - 1)*b^2 - a^2)*ch^2 - 3*a^2 + 1" ] }, "execution_count": 68, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Delta3_4 = expand_ab((Delta_abc)^3, 4)\n", "Delta3_4" ] }, { "cell_type": "code", "execution_count": 69, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}15 \\, {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + 15 \\, a^{4} - 6 \\, {\\left(5 \\, a^{4} - {\\left(5 \\, a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 6 \\, a^{2} + 1$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}15 \\, {\\left(a^{4} - 2 \\, a^{2} b^{2} + b^{4}\\right)} {\\chi}^{4} + 15 \\, a^{4} - 6 \\, {\\left(5 \\, a^{4} - {\\left(5 \\, a^{2} - 1\\right)} b^{2} - a^{2}\\right)} {\\chi}^{2} - 6 \\, a^{2} + 1$$" ], "text/plain": [ "15*(a^4 - 2*a^2*b^2 + b^4)*ch^4 + 15*a^4 - 6*(5*a^4 - (5*a^2 - 1)*b^2 - a^2)*ch^2 - 6*a^2 + 1" ] }, "execution_count": 69, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Delta6_4 = expand_ab((Delta_abc)^6, 4)\n", "Delta6_4" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Expanding at fourth order in $a$ and $b$ (will be required latter):" ] }, { "cell_type": "code", "execution_count": 70, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg_4 = expand_ab(detg, 4)\n", "detg_4 = detg_4.subs({Delta^3: Delta3_4, Delta^6: Delta6_4})\n", "detg_4 = detg_4.simplify_full()\n", "detg_4 = expand_ab(detg_4, 4)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "For the time being, only the expansion at second order in $a$ is required:" ] }, { "cell_type": "code", "execution_count": 71, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{2} - a^{2} + 1\\right)} y^{4} - {\\left({\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{8} + 2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{6} + {\\left(a^{2} {\\chi_0}^{2} - a^{2} - 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m y^{2} + 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left(b^{2} {\\chi_0}^{2} y^{8} + 2 \\, b^{2} {\\chi_0}^{2} y^{6} - 4 \\, b^{2} {\\chi_0}^{2} m y^{2} + 4 \\, b^{2} {\\chi_0}^{2} m^{2} - {\\left(4 \\, b^{2} {\\chi_0}^{2} m - b^{2} {\\chi_0}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 2 \\, {\\left(2 \\, {\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{2} - 2 \\, a^{2} + 1\\right)} m + y^{2}}{y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{2} - a^{2} + 1\\right)} y^{4} - {\\left({\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{8} + 2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{6} + {\\left(a^{2} {\\chi_0}^{2} - a^{2} - 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m y^{2} + 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left(b^{2} {\\chi_0}^{2} y^{8} + 2 \\, b^{2} {\\chi_0}^{2} y^{6} - 4 \\, b^{2} {\\chi_0}^{2} m y^{2} + 4 \\, b^{2} {\\chi_0}^{2} m^{2} - {\\left(4 \\, b^{2} {\\chi_0}^{2} m - b^{2} {\\chi_0}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 2 \\, {\\left(2 \\, {\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{2} - 2 \\, a^{2} + 1\\right)} m + y^{2}}{y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "-(((a^2 - b^2)*ch0^2 - a^2 + 1)*y^4 - ((a^2*ch0^2 - a^2)*y^8 + 2*(a^2*ch0^2 - a^2)*y^6 + (a^2*ch0^2 - a^2 - 4*(a^2*ch0^2 - a^2)*m)*y^4 - 4*(a^2*ch0^2 - a^2)*m*y^2 + 4*(a^2*ch0^2 - a^2)*m^2)*diff(Phi_1(y), y)^2 + (b^2*ch0^2*y^8 + 2*b^2*ch0^2*y^6 - 4*b^2*ch0^2*m*y^2 + 4*b^2*ch0^2*m^2 - (4*b^2*ch0^2*m - b^2*ch0^2)*y^4)*diff(Psi_1(y), y)^2 - 2*(2*(a^2 - b^2)*ch0^2 - 2*a^2 + 1)*m + y^2)/(y^4 + y^2 - 2*m)" ] }, "execution_count": 71, "metadata": { }, "output_type": "execute_result" } ], "source": [ "detg_2 = expand_ab(detg_4, 2)\n", "detg_2 " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The Nambu-Goto Lagrangian at second order in $a$ and $b$:" ] }, { "cell_type": "code", "execution_count": 72, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{2} - a^{2} + 2\\right)} y^{4} - {\\left({\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{8} + 2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{6} + {\\left(a^{2} {\\chi_0}^{2} - a^{2} - 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m y^{2} + 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left(b^{2} {\\chi_0}^{2} y^{8} + 2 \\, b^{2} {\\chi_0}^{2} y^{6} - 4 \\, b^{2} {\\chi_0}^{2} m y^{2} + 4 \\, b^{2} {\\chi_0}^{2} m^{2} - {\\left(4 \\, b^{2} {\\chi_0}^{2} m - b^{2} {\\chi_0}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, {\\left({\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{2} - a^{2} + 1\\right)} m + 2 \\, y^{2}}{2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{2} - a^{2} + 2\\right)} y^{4} - {\\left({\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{8} + 2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{6} + {\\left(a^{2} {\\chi_0}^{2} - a^{2} - 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m y^{2} + 4 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left(b^{2} {\\chi_0}^{2} y^{8} + 2 \\, b^{2} {\\chi_0}^{2} y^{6} - 4 \\, b^{2} {\\chi_0}^{2} m y^{2} + 4 \\, b^{2} {\\chi_0}^{2} m^{2} - {\\left(4 \\, b^{2} {\\chi_0}^{2} m - b^{2} {\\chi_0}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, {\\left({\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{2} - a^{2} + 1\\right)} m + 2 \\, y^{2}}{2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)}}$$" ], "text/plain": [ "1/2*(((a^2 - b^2)*ch0^2 - a^2 + 2)*y^4 - ((a^2*ch0^2 - a^2)*y^8 + 2*(a^2*ch0^2 - a^2)*y^6 + (a^2*ch0^2 - a^2 - 4*(a^2*ch0^2 - a^2)*m)*y^4 - 4*(a^2*ch0^2 - a^2)*m*y^2 + 4*(a^2*ch0^2 - a^2)*m^2)*diff(Phi_1(y), y)^2 + (b^2*ch0^2*y^8 + 2*b^2*ch0^2*y^6 - 4*b^2*ch0^2*m*y^2 + 4*b^2*ch0^2*m^2 - (4*b^2*ch0^2*m - b^2*ch0^2)*y^4)*diff(Psi_1(y), y)^2 - 4*((a^2 - b^2)*ch0^2 - a^2 + 1)*m + 2*y^2)/(y^4 + y^2 - 2*m)" ] }, "execution_count": 72, "metadata": { }, "output_type": "execute_result" } ], "source": [ "L_2 = expand_ab(sqrt(-detg_2), 2)\n", "L_2" ] }, { "cell_type": "code", "execution_count": 73, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-a^{2} {\\chi_0}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + b^{2} {\\chi_0}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 2 \\, a^{2} {\\chi_0}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + a^{2} y^{8} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 2 \\, b^{2} {\\chi_0}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 4 \\, a^{2} {\\chi_0}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, b^{2} {\\chi_0}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - a^{2} {\\chi_0}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 2 \\, a^{2} y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + b^{2} {\\chi_0}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 4 \\, a^{2} {\\chi_0}^{2} m y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, b^{2} {\\chi_0}^{2} m y^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + a^{2} {\\chi_0}^{2} y^{4} - b^{2} {\\chi_0}^{2} y^{4} - 4 \\, a^{2} {\\chi_0}^{2} m^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + a^{2} y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 4 \\, b^{2} {\\chi_0}^{2} m^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} m y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - a^{2} y^{4} + 4 \\, a^{2} m^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} {\\chi_0}^{2} m + 4 \\, b^{2} {\\chi_0}^{2} m + 2 \\, y^{4} + 4 \\, a^{2} m + 2 \\, y^{2} - 4 \\, m$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-a^{2} {\\chi_0}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + b^{2} {\\chi_0}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 2 \\, a^{2} {\\chi_0}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + a^{2} y^{8} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 2 \\, b^{2} {\\chi_0}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 4 \\, a^{2} {\\chi_0}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, b^{2} {\\chi_0}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - a^{2} {\\chi_0}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 2 \\, a^{2} y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + b^{2} {\\chi_0}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 4 \\, a^{2} {\\chi_0}^{2} m y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, b^{2} {\\chi_0}^{2} m y^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + a^{2} {\\chi_0}^{2} y^{4} - b^{2} {\\chi_0}^{2} y^{4} - 4 \\, a^{2} {\\chi_0}^{2} m^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + a^{2} y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 4 \\, b^{2} {\\chi_0}^{2} m^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} m y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - a^{2} y^{4} + 4 \\, a^{2} m^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} {\\chi_0}^{2} m + 4 \\, b^{2} {\\chi_0}^{2} m + 2 \\, y^{4} + 4 \\, a^{2} m + 2 \\, y^{2} - 4 \\, m$$" ], "text/plain": [ "-a^2*ch0^2*y^8*diff(Phi_1(y), y)^2 + b^2*ch0^2*y^8*diff(Psi_1(y), y)^2 - 2*a^2*ch0^2*y^6*diff(Phi_1(y), y)^2 + a^2*y^8*diff(Phi_1(y), y)^2 + 2*b^2*ch0^2*y^6*diff(Psi_1(y), y)^2 + 4*a^2*ch0^2*m*y^4*diff(Phi_1(y), y)^2 - 4*b^2*ch0^2*m*y^4*diff(Psi_1(y), y)^2 - a^2*ch0^2*y^4*diff(Phi_1(y), y)^2 + 2*a^2*y^6*diff(Phi_1(y), y)^2 + b^2*ch0^2*y^4*diff(Psi_1(y), y)^2 + 4*a^2*ch0^2*m*y^2*diff(Phi_1(y), y)^2 - 4*a^2*m*y^4*diff(Phi_1(y), y)^2 - 4*b^2*ch0^2*m*y^2*diff(Psi_1(y), y)^2 + a^2*ch0^2*y^4 - b^2*ch0^2*y^4 - 4*a^2*ch0^2*m^2*diff(Phi_1(y), y)^2 + a^2*y^4*diff(Phi_1(y), y)^2 + 4*b^2*ch0^2*m^2*diff(Psi_1(y), y)^2 - 4*a^2*m*y^2*diff(Phi_1(y), y)^2 - a^2*y^4 + 4*a^2*m^2*diff(Phi_1(y), y)^2 - 4*a^2*ch0^2*m + 4*b^2*ch0^2*m + 2*y^4 + 4*a^2*m + 2*y^2 - 4*m" ] }, "execution_count": 73, "metadata": { }, "output_type": "execute_result" } ], "source": [ "L_2.numerator()" ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, y^{4} + 2 \\, y^{2} - 4 \\, m$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, y^{4} + 2 \\, y^{2} - 4 \\, m$$" ], "text/plain": [ "2*y^4 + 2*y^2 - 4*m" ] }, "execution_count": 74, "metadata": { }, "output_type": "execute_result" } ], "source": [ "L_2.denominator()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Euler-Lagrange equations" ] }, { "cell_type": "code", "execution_count": 75, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def euler_lagrange(lagr, qs, var):\n", " r\"\"\"\n", " Derive the Euler-Lagrange equations from a given Lagrangian.\n", "\n", " INPUT:\n", "\n", " - ``lagr`` -- symbolic expression representing the Lagrangian density\n", " - ``qs`` -- either a single symbolic function or a list/tuple of\n", " symbolic functions, representing the `q`'s; these functions must\n", " appear in ``lagr`` up to at most their first derivatives\n", " - ``var`` -- either a single variable, typically `t` (1-dimensional\n", " problem) or a list/tuple of symbolic variables\n", "\n", " OUTPUT:\n", "\n", " - list of Euler-Lagrange equations; if only one function is involved, the\n", " single Euler-Lagrannge equation is returned instead.\n", "\n", " \"\"\"\n", " if not isinstance(qs, (list, tuple)):\n", " qs = [qs]\n", " if not isinstance(var, (list, tuple)):\n", " var = [var]\n", " n = len(qs)\n", " d = len(var)\n", " qv = [SR.var('qxxxx{}'.format(q)) for q in qs]\n", " dqv = [[SR.var('qxxxx{}_{}'.format(q, v)) for v in var] for q in qs]\n", " subs = {qs[i](*var): qv[i] for i in range(n)}\n", " subs_inv = {qv[i]: qs[i](*var) for i in range(n)}\n", " for i in range(n):\n", " subs.update({diff(qs[i](*var), var[j]): dqv[i][j]\n", " for j in range(d)})\n", " subs_inv.update({dqv[i][j]: diff(qs[i](*var), var[j])\n", " for j in range(d)})\n", " lg = lagr.substitute(subs)\n", " eqs = []\n", " for i in range(n):\n", " dLdq = diff(lg, qv[i]).simplify_full()\n", " dLdq = dLdq.substitute(subs_inv)\n", " ddL = 0\n", " for j in range(d):\n", " h = diff(lg, dqv[i][j]).simplify_full()\n", " h = h.substitute(subs_inv)\n", " ddL += diff(h, var[j])\n", " eqs.append((dLdq - ddL).simplify_full() == 0)\n", " if n == 1:\n", " return eqs[0]\n", " return eqs" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We compute the Euler-Lagrange equations from $L_2$ for $\\Phi_1$ and $\\Psi_1$:" ] }, { "cell_type": "code", "execution_count": 76, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, {\\left(2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{3} + {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{4} + {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0, -2 \\, {\\left(2 \\, b^{2} {\\chi_0}^{2} y^{3} + b^{2} {\\chi_0}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left(b^{2} {\\chi_0}^{2} y^{4} + b^{2} {\\chi_0}^{2} y^{2} - 2 \\, b^{2} {\\chi_0}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0\\right]$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, {\\left(2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{3} + {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{4} + {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0, -2 \\, {\\left(2 \\, b^{2} {\\chi_0}^{2} y^{3} + b^{2} {\\chi_0}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left(b^{2} {\\chi_0}^{2} y^{4} + b^{2} {\\chi_0}^{2} y^{2} - 2 \\, b^{2} {\\chi_0}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0\\right]$$" ], "text/plain": [ "[2*(2*(a^2*ch0^2 - a^2)*y^3 + (a^2*ch0^2 - a^2)*y)*diff(Phi_1(y), y) + ((a^2*ch0^2 - a^2)*y^4 + (a^2*ch0^2 - a^2)*y^2 - 2*(a^2*ch0^2 - a^2)*m)*diff(Phi_1(y), y, y) == 0,\n", " -2*(2*b^2*ch0^2*y^3 + b^2*ch0^2*y)*diff(Psi_1(y), y) - (b^2*ch0^2*y^4 + b^2*ch0^2*y^2 - 2*b^2*ch0^2*m)*diff(Psi_1(y), y, y) == 0]" ] }, "execution_count": 76, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eqs = euler_lagrange(L_2, [Phi_1, Psi_1], y)\n", "eqs" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#### Solving the equation for $\\Phi_1$ (Eq. (4.29))" ] }, { "cell_type": "code", "execution_count": 77, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{3} + {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{4} + {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{3} + {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{4} + {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "2*(2*(a^2*ch0^2 - a^2)*y^3 + (a^2*ch0^2 - a^2)*y)*diff(Phi_1(y), y) + ((a^2*ch0^2 - a^2)*y^4 + (a^2*ch0^2 - a^2)*y^2 - 2*(a^2*ch0^2 - a^2)*m)*diff(Phi_1(y), y, y) == 0" ] }, "execution_count": 77, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_Phi1 = eqs[0]\n", "eq_Phi1" ] }, { "cell_type": "code", "execution_count": 78, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, y^{3} + y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left(y^{4} + y^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, y^{3} + y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left(y^{4} + y^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "2*(2*y^3 + y)*diff(Phi_1(y), y) + (y^4 + y^2 - 2*m)*diff(Phi_1(y), y, y) == 0" ] }, "execution_count": 78, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_Phi1 = (eq_Phi1/(a^2*(ch0^2-1))).simplify_full()\n", "eq_Phi1" ] }, { "cell_type": "code", "execution_count": 79, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{y^{4} + y^{2} - 2 \\, m}\\,{d y} + K_{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{y^{4} + y^{2} - 2 \\, m}\\,{d y} + K_{2}$$" ], "text/plain": [ "_K1*integrate(1/(y^4 + y^2 - 2*m), y) + _K2" ] }, "execution_count": 79, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Phi1_sol(y) = desolve(eq_Phi1, Phi_1(y), ivar=y)\n", "Phi1_sol(y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We recover Eqs. (4.29) with $K_1 = \\mathfrak{p}$ and $K_2=0$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The symbolic constants $K_1$ and $K_2$ are actually denoted `_K1` and `_K2` by SageMath, as `print` reveals:" ] }, { "cell_type": "code", "execution_count": 80, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "_K1*integrate(1/(y^4 + y^2 - 2*m), y) + _K2\n" ] } ], "source": [ "print(Phi1_sol(y))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Hence we perform the substitutions with `SR.var('_K1')` and `SR.var('_K2')`:" ] }, { "cell_type": "code", "execution_count": 81, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mathfrak{p}} \\int \\frac{1}{y^{4} + y^{2} - 2 \\, m}\\,{d y}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mathfrak{p}} \\int \\frac{1}{y^{4} + y^{2} - 2 \\, m}\\,{d y}$$" ], "text/plain": [ "pf*integrate(1/(y^4 + y^2 - 2*m), y)" ] }, "execution_count": 81, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pf = var(\"pf\", latex_name=r\"\\mathfrak{p}\", domain='real')\n", "Phi1_sol(y) = Phi1_sol(y).subs({SR.var('_K1'): pf, SR.var('_K2'): 0})\n", "Phi1_sol(y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#### Solving the equation for $\\Psi_1$ (Eq. (4.29))" ] }, { "cell_type": "code", "execution_count": 82, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, b^{2} {\\chi_0}^{2} y^{3} + b^{2} {\\chi_0}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left(b^{2} {\\chi_0}^{2} y^{4} + b^{2} {\\chi_0}^{2} y^{2} - 2 \\, b^{2} {\\chi_0}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, b^{2} {\\chi_0}^{2} y^{3} + b^{2} {\\chi_0}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left(b^{2} {\\chi_0}^{2} y^{4} + b^{2} {\\chi_0}^{2} y^{2} - 2 \\, b^{2} {\\chi_0}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "-2*(2*b^2*ch0^2*y^3 + b^2*ch0^2*y)*diff(Psi_1(y), y) - (b^2*ch0^2*y^4 + b^2*ch0^2*y^2 - 2*b^2*ch0^2*m)*diff(Psi_1(y), y, y) == 0" ] }, "execution_count": 82, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_Psi1 = eqs[1]\n", "eq_Psi1" ] }, { "cell_type": "code", "execution_count": 83, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, y^{3} + y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left(y^{4} + y^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, y^{3} + y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left(y^{4} + y^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "-2*(2*y^3 + y)*diff(Psi_1(y), y) - (y^4 + y^2 - 2*m)*diff(Psi_1(y), y, y) == 0" ] }, "execution_count": 83, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_Phi1 = (eq_Psi1/(b^2*ch0^2)).simplify_full()\n", "eq_Phi1" ] }, { "cell_type": "code", "execution_count": 84, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{y^{4} + y^{2} - 2 \\, m}\\,{d y} + K_{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{y^{4} + y^{2} - 2 \\, m}\\,{d y} + K_{2}$$" ], "text/plain": [ "_K1*integrate(1/(y^4 + y^2 - 2*m), y) + _K2" ] }, "execution_count": 84, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Psi1_sol(y) = desolve(eq_Psi1, Psi_1(y), ivar=y)\n", "Psi1_sol(y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We recover Eq. (4.29) with $K_1 = \\mathfrak{q}$ and $K_2=0$." ] }, { "cell_type": "code", "execution_count": 85, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mathfrak{q}} \\int \\frac{1}{y^{4} + y^{2} - 2 \\, m}\\,{d y}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mathfrak{q}} \\int \\frac{1}{y^{4} + y^{2} - 2 \\, m}\\,{d y}$$" ], "text/plain": [ "qf*integrate(1/(y^4 + y^2 - 2*m), y)" ] }, "execution_count": 85, "metadata": { }, "output_type": "execute_result" } ], "source": [ "qf = var('qf', latex_name=r\"\\mathfrak{q}\", domain='real')\n", "Psi1_sol(y) = Psi1_sol(y).subs({SR.var('_K1'): qf, SR.var('_K2'): 0})\n", "Psi1_sol(y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Nambu-Goto Lagrangian at fourth order in $a$ and $b$" ] }, { "cell_type": "code", "execution_count": 86, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "L_4 = expand_ab(sqrt(-detg_4), 4)" ] }, { "cell_type": "code", "execution_count": 87, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "eqs = euler_lagrange(L_4, [Phi_1, Psi_1, chi_1], y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### The equation for $\\chi_1$" ] }, { "cell_type": "code", "execution_count": 88, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} y^{4} - {\\left({\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} y^{8} + 2 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} y^{6} + {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m\\right)} y^{4} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m y^{2} + 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left({\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} y^{8} + 2 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} y^{6} + {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0} - 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m\\right)} y^{4} - 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m y^{2} + 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} m + 2 \\, {\\left(2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{7} + 3 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{5} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{3} - 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right) + {\\left({\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{8} + 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{6} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y^{2} + 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\chi_{1}\\left(y\\right)}{{\\left({\\chi_0}^{2} - 1\\right)} y^{4} + {\\left({\\chi_0}^{2} - 1\\right)} y^{2} - 2 \\, {\\left({\\chi_0}^{2} - 1\\right)} m} = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} y^{4} - {\\left({\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} y^{8} + 2 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} y^{6} + {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m\\right)} y^{4} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m y^{2} + 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left({\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} y^{8} + 2 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} y^{6} + {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0} - 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m\\right)} y^{4} - 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m y^{2} + 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} m + 2 \\, {\\left(2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{7} + 3 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{5} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{3} - 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right) + {\\left({\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{8} + 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{6} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y^{2} + 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\chi_{1}\\left(y\\right)}{{\\left({\\chi_0}^{2} - 1\\right)} y^{4} + {\\left({\\chi_0}^{2} - 1\\right)} y^{2} - 2 \\, {\\left({\\chi_0}^{2} - 1\\right)} m} = 0$$" ], "text/plain": [ "(((a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0^3 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0)*y^4 - (((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*y^8 + 2*((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*y^6 + ((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0 - 4*((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*m)*y^4 - 4*((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*m*y^2 + 4*((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*m^2)*diff(Phi_1(y), y)^2 + (((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*y^8 + 2*((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*y^6 + ((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0 - 4*((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*m)*y^4 - 4*((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*m*y^2 + 4*((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*m^2)*diff(Psi_1(y), y)^2 - 4*((a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0^3 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0)*m + 2*(2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^7 + 3*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^5 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m)*y^3 - 2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m*y)*diff(chi_1(y), y) + ((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^8 + 2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^6 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m)*y^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m*y^2 + 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m^2)*diff(chi_1(y), y, y))/((ch0^2 - 1)*y^4 + (ch0^2 - 1)*y^2 - 2*(ch0^2 - 1)*m) == 0" ] }, "execution_count": 88, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_chi1 = eqs[2]\n", "eq_chi1" ] }, { "cell_type": "code", "execution_count": 89, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\chi_0}^{2} - 1\\right)} y^{4} + {\\left({\\chi_0}^{2} - 1\\right)} y^{2} - 2 \\, {\\left({\\chi_0}^{2} - 1\\right)} m$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\chi_0}^{2} - 1\\right)} y^{4} + {\\left({\\chi_0}^{2} - 1\\right)} y^{2} - 2 \\, {\\left({\\chi_0}^{2} - 1\\right)} m$$" ], "text/plain": [ "(ch0^2 - 1)*y^4 + (ch0^2 - 1)*y^2 - 2*(ch0^2 - 1)*m" ] }, "execution_count": 89, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_chi1.lhs().denominator().simplify_full()" ] }, { "cell_type": "code", "execution_count": 90, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} y^{4} - {\\left({\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} y^{8} + 2 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} y^{6} + {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m\\right)} y^{4} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m y^{2} + 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left({\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} y^{8} + 2 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} y^{6} + {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0} - 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m\\right)} y^{4} - 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m y^{2} + 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} m + 2 \\, {\\left(2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{7} + 3 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{5} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{3} - 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right) + {\\left({\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{8} + 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{6} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y^{2} + 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\chi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} y^{4} - {\\left({\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} y^{8} + 2 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} y^{6} + {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m\\right)} y^{4} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m y^{2} + 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left({\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} y^{8} + 2 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} y^{6} + {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0} - 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m\\right)} y^{4} - 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m y^{2} + 4 \\, {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} m^{2}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} m + 2 \\, {\\left(2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{7} + 3 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{5} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{3} - 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right) + {\\left({\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{8} + 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{6} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y^{2} + 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\chi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "((a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0^3 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0)*y^4 - (((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*y^8 + 2*((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*y^6 + ((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0 - 4*((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*m)*y^4 - 4*((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*m*y^2 + 4*((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*m^2)*diff(Phi_1(y), y)^2 + (((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*y^8 + 2*((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*y^6 + ((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0 - 4*((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*m)*y^4 - 4*((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*m*y^2 + 4*((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*m^2)*diff(Psi_1(y), y)^2 - 4*((a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0^3 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0)*m + 2*(2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^7 + 3*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^5 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m)*y^3 - 2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m*y)*diff(chi_1(y), y) + ((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^8 + 2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^6 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m)*y^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m*y^2 + 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m^2)*diff(chi_1(y), y, y) == 0" ] }, "execution_count": 90, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_chi1 = eq_chi1.lhs().numerator().simplify_full() == 0\n", "eq_chi1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We plug the solutions obtained previously for $\\Phi_1(y)$ and $\\Psi_1(y)$ in this equation:" ] }, { "cell_type": "code", "execution_count": 91, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} y^{4} - {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} {\\mathfrak{p}}^{2} + {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} {\\mathfrak{q}}^{2} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} m + 2 \\, {\\left(2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{7} + 3 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{5} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{3} - 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right) + {\\left({\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{8} + 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{6} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y^{2} + 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\chi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} y^{4} - {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} {\\mathfrak{p}}^{2} + {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} {\\mathfrak{q}}^{2} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} m + 2 \\, {\\left(2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{7} + 3 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{5} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{3} - 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right) + {\\left({\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{8} + 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{6} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y^{2} + 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\chi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "((a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0^3 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0)*y^4 - ((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*pf^2 + ((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*qf^2 - 4*((a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0^3 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0)*m + 2*(2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^7 + 3*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^5 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m)*y^3 - 2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m*y)*diff(chi_1(y), y) + ((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^8 + 2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^6 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m)*y^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m*y^2 + 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m^2)*diff(chi_1(y), y, y) == 0" ] }, "execution_count": 91, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_chi1 = eq_chi1.substitute_function(Phi_1, \n", " Phi1_sol).substitute_function(Psi_1, \n", " Psi1_sol)\n", "eq_chi1 = eq_chi1.simplify_full()\n", "eq_chi1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (4.30)" ] }, { "cell_type": "code", "execution_count": 92, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} y^{4} - {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} {\\mathfrak{p}}^{2} + {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} {\\mathfrak{q}}^{2} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} m + 2 \\, {\\left(2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{7} + 3 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{5} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{3} - 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right) + {\\left({\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{8} + 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{6} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y^{2} + 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\chi_{1}\\left(y\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} y^{4} - {\\left({\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b + a^{2} b^{2}\\right)} {\\chi_0}\\right)} {\\mathfrak{p}}^{2} + {\\left({\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{2} b^{2} + 2 \\, a b^{3} + b^{4}\\right)} {\\chi_0}\\right)} {\\mathfrak{q}}^{2} - 4 \\, {\\left({\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}^{3} - {\\left(a^{4} + 2 \\, a^{3} b - 2 \\, a b^{3} - b^{4}\\right)} {\\chi_0}\\right)} m + 2 \\, {\\left(2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{7} + 3 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{5} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{3} - 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right) + {\\left({\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{8} + 2 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} y^{6} + {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m y^{2} + 4 \\, {\\left(a^{4} + 4 \\, a^{3} b + 6 \\, a^{2} b^{2} + 4 \\, a b^{3} + b^{4}\\right)} m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\chi_{1}\\left(y\\right)$$" ], "text/plain": [ "((a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0^3 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0)*y^4 - ((a^4 + 2*a^3*b + a^2*b^2)*ch0^3 - (a^4 + 2*a^3*b + a^2*b^2)*ch0)*pf^2 + ((a^2*b^2 + 2*a*b^3 + b^4)*ch0^3 - (a^2*b^2 + 2*a*b^3 + b^4)*ch0)*qf^2 - 4*((a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0^3 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*ch0)*m + 2*(2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^7 + 3*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^5 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m)*y^3 - 2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m*y)*diff(chi_1(y), y) + ((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^8 + 2*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*y^6 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m)*y^4 - 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m*y^2 + 4*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*m^2)*diff(chi_1(y), y, y)" ] }, "execution_count": 92, "metadata": { }, "output_type": "execute_result" } ], "source": [ "lhs = eq_chi1.lhs()\n", "lhs = lhs.simplify_full()\n", "lhs" ] }, { "cell_type": "code", "execution_count": 93, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(y^{4} + y^{2} - 2 \\, m\\right)}^{2} {\\left(a + b\\right)}^{4}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(y^{4} + y^{2} - 2 \\, m\\right)}^{2} {\\left(a + b\\right)}^{4}$$" ], "text/plain": [ "(y^4 + y^2 - 2*m)^2*(a + b)^4" ] }, "execution_count": 93, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = lhs.coefficient(diff(chi_1(y), y, 2)) # coefficient of chi_1''\n", "s.factor()" ] }, { "cell_type": "code", "execution_count": 94, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{2} - b^{2}\\right)} {\\chi_0}\\right)} y^{4} - {\\left(a^{2} {\\chi_0}^{3} - a^{2} {\\chi_0}\\right)} {\\mathfrak{p}}^{2} + {\\left(b^{2} {\\chi_0}^{3} - b^{2} {\\chi_0}\\right)} {\\mathfrak{q}}^{2} - 4 \\, {\\left({\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{2} - b^{2}\\right)} {\\chi_0}\\right)} m + 2 \\, {\\left(2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{7} + 3 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{5} + {\\left(a^{2} + 2 \\, a b + b^{2} - 4 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} y^{3} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m y\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right)}{{\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{8} + 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{6} + {\\left(a^{2} + 2 \\, a b + b^{2} - 4 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m y^{2} + 4 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{2} - b^{2}\\right)} {\\chi_0}\\right)} y^{4} - {\\left(a^{2} {\\chi_0}^{3} - a^{2} {\\chi_0}\\right)} {\\mathfrak{p}}^{2} + {\\left(b^{2} {\\chi_0}^{3} - b^{2} {\\chi_0}\\right)} {\\mathfrak{q}}^{2} - 4 \\, {\\left({\\left(a^{2} - b^{2}\\right)} {\\chi_0}^{3} - {\\left(a^{2} - b^{2}\\right)} {\\chi_0}\\right)} m + 2 \\, {\\left(2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{7} + 3 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{5} + {\\left(a^{2} + 2 \\, a b + b^{2} - 4 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} y^{3} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m y\\right)} \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right)}{{\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{8} + 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{6} + {\\left(a^{2} + 2 \\, a b + b^{2} - 4 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} y^{4} - 4 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m y^{2} + 4 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m^{2}}$$" ], "text/plain": [ "(((a^2 - b^2)*ch0^3 - (a^2 - b^2)*ch0)*y^4 - (a^2*ch0^3 - a^2*ch0)*pf^2 + (b^2*ch0^3 - b^2*ch0)*qf^2 - 4*((a^2 - b^2)*ch0^3 - (a^2 - b^2)*ch0)*m + 2*(2*(a^2 + 2*a*b + b^2)*y^7 + 3*(a^2 + 2*a*b + b^2)*y^5 + (a^2 + 2*a*b + b^2 - 4*(a^2 + 2*a*b + b^2)*m)*y^3 - 2*(a^2 + 2*a*b + b^2)*m*y)*diff(chi_1(y), y))/((a^2 + 2*a*b + b^2)*y^8 + 2*(a^2 + 2*a*b + b^2)*y^6 + (a^2 + 2*a*b + b^2 - 4*(a^2 + 2*a*b + b^2)*m)*y^4 - 4*(a^2 + 2*a*b + b^2)*m*y^2 + 4*(a^2 + 2*a*b + b^2)*m^2)" ] }, "execution_count": 94, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = (lhs/s - diff(chi_1(y), y, 2)).simplify_full()\n", "s1" ] }, { "cell_type": "code", "execution_count": 95, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, y^{2} + 1\\right)} y}{y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, y^{2} + 1\\right)} y}{y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "2*(2*y^2 + 1)*y/(y^4 + y^2 - 2*m)" ] }, "execution_count": 95, "metadata": { }, "output_type": "execute_result" } ], "source": [ "b1 = s1.coefficient(diff(chi_1(y), y)).factor()\n", "b1" ] }, { "cell_type": "code", "execution_count": 96, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(a^{2} y^{4} - b^{2} y^{4} - a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} - 4 \\, a^{2} m + 4 \\, b^{2} m\\right)} {\\left({\\chi_0} + 1\\right)} {\\left({\\chi_0} - 1\\right)} {\\chi_0}}{{\\left(y^{4} + y^{2} - 2 \\, m\\right)}^{2} {\\left(a + b\\right)}^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(a^{2} y^{4} - b^{2} y^{4} - a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} - 4 \\, a^{2} m + 4 \\, b^{2} m\\right)} {\\left({\\chi_0} + 1\\right)} {\\left({\\chi_0} - 1\\right)} {\\chi_0}}{{\\left(y^{4} + y^{2} - 2 \\, m\\right)}^{2} {\\left(a + b\\right)}^{2}}$$" ], "text/plain": [ "(a^2*y^4 - b^2*y^4 - a^2*pf^2 + b^2*qf^2 - 4*a^2*m + 4*b^2*m)*(ch0 + 1)*(ch0 - 1)*ch0/((y^4 + y^2 - 2*m)^2*(a + b)^2)" ] }, "execution_count": 96, "metadata": { }, "output_type": "execute_result" } ], "source": [ "b2 = (s1 - b1*diff(chi_1(y), y)).simplify_full().factor()\n", "b2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The equation for $\\chi_1$ is thus:" ] }, { "cell_type": "code", "execution_count": 97, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, y^{2} + 1\\right)} y \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right)}{y^{4} + y^{2} - 2 \\, m} + \\frac{{\\left(a^{2} y^{4} - b^{2} y^{4} - a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} - 4 \\, a^{2} m + 4 \\, b^{2} m\\right)} {\\left({\\chi_0} + 1\\right)} {\\left({\\chi_0} - 1\\right)} {\\chi_0}}{{\\left(y^{4} + y^{2} - 2 \\, m\\right)}^{2} {\\left(a + b\\right)}^{2}} + \\frac{\\partial^{2}}{(\\partial y)^{2}}\\chi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, y^{2} + 1\\right)} y \\frac{\\partial}{\\partial y}\\chi_{1}\\left(y\\right)}{y^{4} + y^{2} - 2 \\, m} + \\frac{{\\left(a^{2} y^{4} - b^{2} y^{4} - a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} - 4 \\, a^{2} m + 4 \\, b^{2} m\\right)} {\\left({\\chi_0} + 1\\right)} {\\left({\\chi_0} - 1\\right)} {\\chi_0}}{{\\left(y^{4} + y^{2} - 2 \\, m\\right)}^{2} {\\left(a + b\\right)}^{2}} + \\frac{\\partial^{2}}{(\\partial y)^{2}}\\chi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "2*(2*y^2 + 1)*y*diff(chi_1(y), y)/(y^4 + y^2 - 2*m) + (a^2*y^4 - b^2*y^4 - a^2*pf^2 + b^2*qf^2 - 4*a^2*m + 4*b^2*m)*(ch0 + 1)*(ch0 - 1)*ch0/((y^4 + y^2 - 2*m)^2*(a + b)^2) + diff(chi_1(y), y, y) == 0" ] }, "execution_count": 97, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_chi1 = diff(chi_1(y), y, 2) + b1*diff(chi_1(y), y) + b2 == 0\n", "eq_chi1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Given that \n", "$$ \\chi_1(y) = - \\sin\\Theta_0 \\; \\Theta_1(y) = - \\sqrt{1-\\chi_0^2} \\; \\Theta_1(y)$$\n", "and\n", "$$\\sin2\\Theta_0 = 2\\chi_0\\sqrt{1-\\chi_0^2}$$\n", "we get for the following equation for $\\Upsilon = \\Theta_1'$ \n", "(defining $\\Theta_2 := 2 \\Theta_0$):" ] }, { "cell_type": "code", "execution_count": 98, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, y^{2} + 1\\right)} y \\Upsilon\\left(y\\right)}{y^{4} + y^{2} - 2 \\, m} + \\frac{{\\left(a^{2} y^{4} - b^{2} y^{4} - a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} - 4 \\, a^{2} m + 4 \\, b^{2} m\\right)} \\sin\\left({\\Theta_2}\\right)}{2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)}^{2} {\\left(a + b\\right)}^{2}} + \\frac{\\partial}{\\partial y}\\Upsilon\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, y^{2} + 1\\right)} y \\Upsilon\\left(y\\right)}{y^{4} + y^{2} - 2 \\, m} + \\frac{{\\left(a^{2} y^{4} - b^{2} y^{4} - a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} - 4 \\, a^{2} m + 4 \\, b^{2} m\\right)} \\sin\\left({\\Theta_2}\\right)}{2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)}^{2} {\\left(a + b\\right)}^{2}} + \\frac{\\partial}{\\partial y}\\Upsilon\\left(y\\right) = 0$$" ], "text/plain": [ "2*(2*y^2 + 1)*y*Y(y)/(y^4 + y^2 - 2*m) + 1/2*(a^2*y^4 - b^2*y^4 - a^2*pf^2 + b^2*qf^2 - 4*a^2*m + 4*b^2*m)*sin(Th2)/((y^4 + y^2 - 2*m)^2*(a + b)^2) + diff(Y(y), y) == 0" ] }, "execution_count": 98, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y = function('Y', latex_name=r'\\Upsilon')\n", "Th2 = var('Th2', latex_name=r'\\Theta_2', domain='real')\n", "eq_Y = diff(Y(y), y) + b1*Y(y) \\\n", " - b2/(1 - ch0)/(1 + ch0)/ch0*sin(Th2)/2 == 0\n", "eq_Y" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This agrees with Eq. (4.30) of the paper." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Solving the equation for $\\Upsilon := \\Theta_1'$" ] }, { "cell_type": "code", "execution_count": 99, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "Y_sol(y) = desolve(eq_Y, Y(y), ivar=y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The solution involves an integral that SageMath is not capable to evaluate with the default integrator. Printing `Y_sol` provides the unvaluated form of the integral, in order to compute it by means of FriCAS:" ] }, { "cell_type": "code", "execution_count": 100, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1/2*(2*_C - (a*sin(Th2) - b*sin(Th2))*y/(a + b) - integrate(-(a^2*pf^2 - b^2*qf^2 + (a^2 - b^2)*y^2 + 2*(a^2 - b^2)*m)/(y^4 + y^2 - 2*m), y)*sin(Th2)/(a^2 + 2*a*b + b^2))/(y^4 + y^2 - 2*m)\n" ] } ], "source": [ "print(Y_sol(y))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The solution involves some constant, denoted `_C` by SageMath. We rename it `C_1` and \n", "rewrite the above solution as" ] }, { "cell_type": "code", "execution_count": 101, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, C_{1} - \\frac{{\\left(a \\sin\\left({\\Theta_2}\\right) - b \\sin\\left({\\Theta_2}\\right)\\right)} y}{a + b} - \\frac{{\\rm Integ}\\left(y\\right) \\sin\\left({\\Theta_2}\\right)}{a^{2} + 2 \\, a b + b^{2}}}{2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, C_{1} - \\frac{{\\left(a \\sin\\left({\\Theta_2}\\right) - b \\sin\\left({\\Theta_2}\\right)\\right)} y}{a + b} - \\frac{{\\rm Integ}\\left(y\\right) \\sin\\left({\\Theta_2}\\right)}{a^{2} + 2 \\, a b + b^{2}}}{2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)}}$$" ], "text/plain": [ "1/2*(2*C_1 - (a*sin(Th2) - b*sin(Th2))*y/(a + b) - Integ(y)*sin(Th2)/(a^2 + 2*a*b + b^2))/(y^4 + y^2 - 2*m)" ] }, "execution_count": 101, "metadata": { }, "output_type": "execute_result" } ], "source": [ "C_1 = var('C_1')\n", "Integ(y) = function('Integ')(y)\n", "Y_sol0(y) = 1/2*(2*C_1 - (a*sin(Th2) - b*sin(Th2))*y/(a + b) \\\n", " - Integ(y)*sin(Th2)/(a^2 + 2*a*b + b^2))/(y^4 + y^2 - 2*m)\n", "Y_sol0(y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "`Integ(y)` represents the integral $I(y)$, whose integrand, $F(y)$ say, is read from the\n", "output of `print(Y_sol(Y))`:" ] }, { "cell_type": "code", "execution_count": 102, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + {\\left(a^{2} - b^{2}\\right)} y^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m}{y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + {\\left(a^{2} - b^{2}\\right)} y^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m}{y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "-(a^2*pf^2 - b^2*qf^2 + (a^2 - b^2)*y^2 + 2*(a^2 - b^2)*m)/(y^4 + y^2 - 2*m)" ] }, "execution_count": 102, "metadata": { }, "output_type": "execute_result" } ], "source": [ "F(y) = -(a^2*pf^2 - b^2*qf^2 + (a^2 - b^2)*y^2 + 2*(a^2 - b^2)*m)/(y^4 + y^2 - 2*m)\n", "F(y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We split the integral in two parts:\n", "$$ I(y) = F_1 \\; s_1(y) + F_2 \\; s_2(y)$$\n", "with \n", "$$ s_1(y) := \\int^y \\frac{\\bar{y}^2}{\\bar{y}^4 + \\bar{y}^2 - 2m} \\, \\mathrm{d}\\bar{y}, \\qquad s_2(y) := \\int^y \\frac{\\mathrm{d}\\bar{y}}{\\bar{y}^4 + \\bar{y}^2 - 2m} $$\n", "and" ] }, { "cell_type": "code", "execution_count": 103, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-a^{2} + b^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-a^{2} + b^{2}$$" ], "text/plain": [ "-a^2 + b^2" ] }, "execution_count": 103, "metadata": { }, "output_type": "execute_result" } ], "source": [ "F1 = -(a^2 - b^2)\n", "F1" ] }, { "cell_type": "code", "execution_count": 104, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} - 2 \\, {\\left(a^{2} - b^{2}\\right)} m$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} - 2 \\, {\\left(a^{2} - b^{2}\\right)} m$$" ], "text/plain": [ "-a^2*pf^2 + b^2*qf^2 - 2*(a^2 - b^2)*m" ] }, "execution_count": 104, "metadata": { }, "output_type": "execute_result" } ], "source": [ "F2 = -(a^2*pf^2 - b^2*qf^2 + 2*(a^2 - b^2)*m)\n", "F2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check:" ] }, { "cell_type": "code", "execution_count": 105, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 105, "metadata": { }, "output_type": "execute_result" } ], "source": [ "bool(F(y) == F1*y^2/(y^4 + y^2 - 2*m) + F2/(y^4 + y^2 - 2*m))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us evaluate $s_1(y)$ by means of FriCAS:" ] }, { "cell_type": "code", "execution_count": 106, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\sqrt{8 \\, m + 1} + 1}{8 \\, m + 1}} \\log\\left(\\sqrt{\\frac{1}{2}} \\sqrt{8 \\, m + 1} \\sqrt{-\\frac{\\sqrt{8 \\, m + 1} + 1}{8 \\, m + 1}} + y\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\sqrt{8 \\, m + 1} + 1}{8 \\, m + 1}} \\log\\left(-\\sqrt{\\frac{1}{2}} \\sqrt{8 \\, m + 1} \\sqrt{-\\frac{\\sqrt{8 \\, m + 1} + 1}{8 \\, m + 1}} + y\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\sqrt{8 \\, m + 1} - 1}{8 \\, m + 1}} \\log\\left(\\sqrt{\\frac{1}{2}} \\sqrt{8 \\, m + 1} \\sqrt{\\frac{\\sqrt{8 \\, m + 1} - 1}{8 \\, m + 1}} + y\\right) + \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\sqrt{8 \\, m + 1} - 1}{8 \\, m + 1}} \\log\\left(-\\sqrt{\\frac{1}{2}} \\sqrt{8 \\, m + 1} \\sqrt{\\frac{\\sqrt{8 \\, m + 1} - 1}{8 \\, m + 1}} + y\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\sqrt{8 \\, m + 1} + 1}{8 \\, m + 1}} \\log\\left(\\sqrt{\\frac{1}{2}} \\sqrt{8 \\, m + 1} \\sqrt{-\\frac{\\sqrt{8 \\, m + 1} + 1}{8 \\, m + 1}} + y\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\sqrt{8 \\, m + 1} + 1}{8 \\, m + 1}} \\log\\left(-\\sqrt{\\frac{1}{2}} \\sqrt{8 \\, m + 1} \\sqrt{-\\frac{\\sqrt{8 \\, m + 1} + 1}{8 \\, m + 1}} + y\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\sqrt{8 \\, m + 1} - 1}{8 \\, m + 1}} \\log\\left(\\sqrt{\\frac{1}{2}} \\sqrt{8 \\, m + 1} \\sqrt{\\frac{\\sqrt{8 \\, m + 1} - 1}{8 \\, m + 1}} + y\\right) + \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\sqrt{8 \\, m + 1} - 1}{8 \\, m + 1}} \\log\\left(-\\sqrt{\\frac{1}{2}} \\sqrt{8 \\, m + 1} \\sqrt{\\frac{\\sqrt{8 \\, m + 1} - 1}{8 \\, m + 1}} + y\\right)$$" ], "text/plain": [ "1/2*sqrt(1/2)*sqrt(-(sqrt(8*m + 1) + 1)/(8*m + 1))*log(sqrt(1/2)*sqrt(8*m + 1)*sqrt(-(sqrt(8*m + 1) + 1)/(8*m + 1)) + y) - 1/2*sqrt(1/2)*sqrt(-(sqrt(8*m + 1) + 1)/(8*m + 1))*log(-sqrt(1/2)*sqrt(8*m + 1)*sqrt(-(sqrt(8*m + 1) + 1)/(8*m + 1)) + y) - 1/2*sqrt(1/2)*sqrt((sqrt(8*m + 1) - 1)/(8*m + 1))*log(sqrt(1/2)*sqrt(8*m + 1)*sqrt((sqrt(8*m + 1) - 1)/(8*m + 1)) + y) + 1/2*sqrt(1/2)*sqrt((sqrt(8*m + 1) - 1)/(8*m + 1))*log(-sqrt(1/2)*sqrt(8*m + 1)*sqrt((sqrt(8*m + 1) - 1)/(8*m + 1)) + y)" ] }, "execution_count": 106, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = integrate(y^2/(y^4 + y^2 - 2*m), y, algorithm='fricas')\n", "s1" ] }, { "cell_type": "code", "execution_count": 107, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{\\sqrt{8 \\, m + 1} - 1} \\log\\left(\\frac{\\sqrt{2} y - \\sqrt{\\sqrt{8 \\, m + 1} - 1}}{\\sqrt{2} y + \\sqrt{\\sqrt{8 \\, m + 1} - 1}}\\right) + \\sqrt{2} \\sqrt{-\\sqrt{8 \\, m + 1} - 1} \\log\\left(\\frac{\\sqrt{2} y + \\sqrt{-\\sqrt{8 \\, m + 1} - 1}}{\\sqrt{2} y - \\sqrt{-\\sqrt{8 \\, m + 1} - 1}}\\right)}{4 \\, \\sqrt{8 \\, m + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{\\sqrt{8 \\, m + 1} - 1} \\log\\left(\\frac{\\sqrt{2} y - \\sqrt{\\sqrt{8 \\, m + 1} - 1}}{\\sqrt{2} y + \\sqrt{\\sqrt{8 \\, m + 1} - 1}}\\right) + \\sqrt{2} \\sqrt{-\\sqrt{8 \\, m + 1} - 1} \\log\\left(\\frac{\\sqrt{2} y + \\sqrt{-\\sqrt{8 \\, m + 1} - 1}}{\\sqrt{2} y - \\sqrt{-\\sqrt{8 \\, m + 1} - 1}}\\right)}{4 \\, \\sqrt{8 \\, m + 1}}$$" ], "text/plain": [ "1/4*(sqrt(2)*sqrt(sqrt(8*m + 1) - 1)*log((sqrt(2)*y - sqrt(sqrt(8*m + 1) - 1))/(sqrt(2)*y + sqrt(sqrt(8*m + 1) - 1))) + sqrt(2)*sqrt(-sqrt(8*m + 1) - 1)*log((sqrt(2)*y + sqrt(-sqrt(8*m + 1) - 1))/(sqrt(2)*y - sqrt(-sqrt(8*m + 1) - 1))))/sqrt(8*m + 1)" ] }, "execution_count": 107, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = s1.canonicalize_radical().simplify_log()\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check:" ] }, { "cell_type": "code", "execution_count": 108, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "y^2/(y^4 + y^2 - 2*m)" ] }, "execution_count": 108, "metadata": { }, "output_type": "execute_result" } ], "source": [ "diff(s1, y).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Similarly, we evaluate $s_2(y)$ by means of FriCAS:" ] }, { "cell_type": "code", "execution_count": 109, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1}{8 \\, m^{2} + m}} \\log\\left(\\frac{1}{2} \\, {\\left(8 \\, m - \\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1}{8 \\, m^{2} + m}} + 2 \\, y\\right) + \\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1}{8 \\, m^{2} + m}} \\log\\left(-\\frac{1}{2} \\, {\\left(8 \\, m - \\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1}{8 \\, m^{2} + m}} + 2 \\, y\\right) - \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} - 1}{8 \\, m^{2} + m}} \\log\\left(\\frac{1}{2} \\, {\\left(8 \\, m + \\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} - 1}{8 \\, m^{2} + m}} + 2 \\, y\\right) + \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} - 1}{8 \\, m^{2} + m}} \\log\\left(-\\frac{1}{2} \\, {\\left(8 \\, m + \\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} - 1}{8 \\, m^{2} + m}} + 2 \\, y\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1}{8 \\, m^{2} + m}} \\log\\left(\\frac{1}{2} \\, {\\left(8 \\, m - \\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1}{8 \\, m^{2} + m}} + 2 \\, y\\right) + \\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1}{8 \\, m^{2} + m}} \\log\\left(-\\frac{1}{2} \\, {\\left(8 \\, m - \\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1}{8 \\, m^{2} + m}} + 2 \\, y\\right) - \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} - 1}{8 \\, m^{2} + m}} \\log\\left(\\frac{1}{2} \\, {\\left(8 \\, m + \\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} - 1}{8 \\, m^{2} + m}} + 2 \\, y\\right) + \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} - 1}{8 \\, m^{2} + m}} \\log\\left(-\\frac{1}{2} \\, {\\left(8 \\, m + \\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, m^{2} + m}{\\sqrt{8 \\, m^{3} + m^{2}}} - 1}{8 \\, m^{2} + m}} + 2 \\, y\\right)$$" ], "text/plain": [ "-1/4*sqrt(((8*m^2 + m)/sqrt(8*m^3 + m^2) + 1)/(8*m^2 + m))*log(1/2*(8*m - (8*m^2 + m)/sqrt(8*m^3 + m^2) + 1)*sqrt(((8*m^2 + m)/sqrt(8*m^3 + m^2) + 1)/(8*m^2 + m)) + 2*y) + 1/4*sqrt(((8*m^2 + m)/sqrt(8*m^3 + m^2) + 1)/(8*m^2 + m))*log(-1/2*(8*m - (8*m^2 + m)/sqrt(8*m^3 + m^2) + 1)*sqrt(((8*m^2 + m)/sqrt(8*m^3 + m^2) + 1)/(8*m^2 + m)) + 2*y) - 1/4*sqrt(-((8*m^2 + m)/sqrt(8*m^3 + m^2) - 1)/(8*m^2 + m))*log(1/2*(8*m + (8*m^2 + m)/sqrt(8*m^3 + m^2) + 1)*sqrt(-((8*m^2 + m)/sqrt(8*m^3 + m^2) - 1)/(8*m^2 + m)) + 2*y) + 1/4*sqrt(-((8*m^2 + m)/sqrt(8*m^3 + m^2) - 1)/(8*m^2 + m))*log(-1/2*(8*m + (8*m^2 + m)/sqrt(8*m^3 + m^2) + 1)*sqrt(-((8*m^2 + m)/sqrt(8*m^3 + m^2) - 1)/(8*m^2 + m)) + 2*y)" ] }, "execution_count": 109, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = integrate(1/(y^4 + y^2 - 2*m), y, algorithm='fricas')\n", "s2" ] }, { "cell_type": "code", "execution_count": 110, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{-8 \\, m + \\sqrt{8 \\, m + 1} - 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, m + 1\\right)}^{\\frac{1}{4}} \\sqrt{m} y - \\sqrt{-8 \\, m + \\sqrt{8 \\, m + 1} - 1} {\\left(\\sqrt{8 \\, m + 1} + 1\\right)}}{4 \\, {\\left(8 \\, m + 1\\right)}^{\\frac{1}{4}} \\sqrt{m} y + \\sqrt{-8 \\, m + \\sqrt{8 \\, m + 1} - 1} {\\left(\\sqrt{8 \\, m + 1} + 1\\right)}}\\right) + \\sqrt{8 \\, m + \\sqrt{8 \\, m + 1} + 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, m + 1\\right)}^{\\frac{1}{4}} \\sqrt{m} y - \\sqrt{8 \\, m + \\sqrt{8 \\, m + 1} + 1} {\\left(\\sqrt{8 \\, m + 1} - 1\\right)}}{4 \\, {\\left(8 \\, m + 1\\right)}^{\\frac{1}{4}} \\sqrt{m} y + \\sqrt{8 \\, m + \\sqrt{8 \\, m + 1} + 1} {\\left(\\sqrt{8 \\, m + 1} - 1\\right)}}\\right)}{4 \\, {\\left(8 \\, m + 1\\right)}^{\\frac{3}{4}} \\sqrt{m}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{-8 \\, m + \\sqrt{8 \\, m + 1} - 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, m + 1\\right)}^{\\frac{1}{4}} \\sqrt{m} y - \\sqrt{-8 \\, m + \\sqrt{8 \\, m + 1} - 1} {\\left(\\sqrt{8 \\, m + 1} + 1\\right)}}{4 \\, {\\left(8 \\, m + 1\\right)}^{\\frac{1}{4}} \\sqrt{m} y + \\sqrt{-8 \\, m + \\sqrt{8 \\, m + 1} - 1} {\\left(\\sqrt{8 \\, m + 1} + 1\\right)}}\\right) + \\sqrt{8 \\, m + \\sqrt{8 \\, m + 1} + 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, m + 1\\right)}^{\\frac{1}{4}} \\sqrt{m} y - \\sqrt{8 \\, m + \\sqrt{8 \\, m + 1} + 1} {\\left(\\sqrt{8 \\, m + 1} - 1\\right)}}{4 \\, {\\left(8 \\, m + 1\\right)}^{\\frac{1}{4}} \\sqrt{m} y + \\sqrt{8 \\, m + \\sqrt{8 \\, m + 1} + 1} {\\left(\\sqrt{8 \\, m + 1} - 1\\right)}}\\right)}{4 \\, {\\left(8 \\, m + 1\\right)}^{\\frac{3}{4}} \\sqrt{m}}$$" ], "text/plain": [ "1/4*(sqrt(-8*m + sqrt(8*m + 1) - 1)*log((4*(8*m + 1)^(1/4)*sqrt(m)*y - sqrt(-8*m + sqrt(8*m + 1) - 1)*(sqrt(8*m + 1) + 1))/(4*(8*m + 1)^(1/4)*sqrt(m)*y + sqrt(-8*m + sqrt(8*m + 1) - 1)*(sqrt(8*m + 1) + 1))) + sqrt(8*m + sqrt(8*m + 1) + 1)*log((4*(8*m + 1)^(1/4)*sqrt(m)*y - sqrt(8*m + sqrt(8*m + 1) + 1)*(sqrt(8*m + 1) - 1))/(4*(8*m + 1)^(1/4)*sqrt(m)*y + sqrt(8*m + sqrt(8*m + 1) + 1)*(sqrt(8*m + 1) - 1))))/((8*m + 1)^(3/4)*sqrt(m))" ] }, "execution_count": 110, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.canonicalize_radical().simplify_log()\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check:" ] }, { "cell_type": "code", "execution_count": 111, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "1/(y^4 + y^2 - 2*m)" ] }, "execution_count": 111, "metadata": { }, "output_type": "execute_result" } ], "source": [ "diff(s2, y).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In the above expressions for $s_1(y)$ and $s_2(y)$, there appears $\\sqrt{1 + 8 m}$,\n", "which can be rewritten\n", "$$\n", " \\sqrt{1 + 8 m} = 2 y_H^2 + 1 \n", "$$\n", "where $y_H$ is the positive root of $y_H^4 + y_H^2 - 2m = 0$. More precisely, we perform the following substitution:\n", "$$\n", " m = \\frac{1}{2} y_H^2 (y_H^2 + 1)\n", "$$" ] }, { "cell_type": "code", "execution_count": 112, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{y_H} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right) + i \\, \\sqrt{{y_H}^{2} + 1} \\log\\left(\\frac{y - i \\, \\sqrt{{y_H}^{2} + 1}}{y + i \\, \\sqrt{{y_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{y_H} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right) + i \\, \\sqrt{{y_H}^{2} + 1} \\log\\left(\\frac{y - i \\, \\sqrt{{y_H}^{2} + 1}}{y + i \\, \\sqrt{{y_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)}}$$" ], "text/plain": [ "-1/2*(yH*log((y + yH)/(y - yH)) + I*sqrt(yH^2 + 1)*log((y - I*sqrt(yH^2 + 1))/(y + I*sqrt(yH^2 + 1))))/(2*yH^2 + 1)" ] }, "execution_count": 112, "metadata": { }, "output_type": "execute_result" } ], "source": [ "yH = var('yH', latex_name=r'y_H', domain='real')\n", "assume(yH > 0)\n", "m_yH = yH^2*(yH^2 + 1)/2\n", "s1 = s1.subs({m: m_yH}).canonicalize_radical().simplify_log()\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In the second $\\log$, we recognize the $\\mathrm{arccot}$ function, via the identity\n", "$$\n", " \\mathrm{arccot}\\, x = \\frac{i}{2} \\ln\\left( \\frac{x - i}{x + i} \\right) . \n", "$$\n", "Given that $\\mathrm{arccot}\\, x = \\pi/2 - \\mathrm{arctan}\\, x$, we use this identity as\n", "$$\n", "i \\ln\\left( \\frac{x - i}{x + i} \\right) = - 2 \\, \\mathrm{arctan}(x) + \\pi\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Thus, we perform the following substitution, disregarding the additive constant $\\pi$:" ] }, { "cell_type": "code", "execution_count": 113, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{y_H} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right) - 2 \\, \\sqrt{{y_H}^{2} + 1} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{y_H} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right) - 2 \\, \\sqrt{{y_H}^{2} + 1} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)}}$$" ], "text/plain": [ "-1/2*(yH*log((y + yH)/(y - yH)) - 2*sqrt(yH^2 + 1)*arctan(y/sqrt(yH^2 + 1)))/(2*yH^2 + 1)" ] }, "execution_count": 113, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = s1.subs({I*sqrt(yH^2 + 1)*log((y - I*sqrt(yH^2 + 1))/(y + I*sqrt(yH^2 + 1))):\n", " -2*sqrt(yH^2 + 1)*atan(y/sqrt(yH^2 + 1))})\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us check that we have indeed a primitive of $y\\mapsto \\frac{y^2}{y^4 + y^2 - 2m}$:" ] }, { "cell_type": "code", "execution_count": 114, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{y^{4} - {y_H}^{4} + y^{2} - {y_H}^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{y^{4} - {y_H}^{4} + y^{2} - {y_H}^{2}}$$" ], "text/plain": [ "y^2/(y^4 - yH^4 + y^2 - yH^2)" ] }, "execution_count": 114, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Ds1 = diff(s1, y).simplify_full()\n", "Ds1" ] }, { "cell_type": "code", "execution_count": 115, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "y^2/(y^4 + y^2 - 2*m)" ] }, "execution_count": 115, "metadata": { }, "output_type": "execute_result" } ], "source": [ "yH_m = sqrt(sqrt(1 + 8*m) - 1)/sqrt(2)\n", "Ds1.subs({yH: yH_m}).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Similarly, let us express $s_2$ in terms of $y_H$:" ] }, { "cell_type": "code", "execution_count": 116, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{i \\, {y_H} \\log\\left(\\frac{-i \\, {y_H}^{2} + \\sqrt{{y_H}^{2} + 1} y - i}{i \\, {y_H}^{2} + \\sqrt{{y_H}^{2} + 1} y + i}\\right) + \\sqrt{{y_H}^{2} + 1} \\log\\left(\\frac{y - {y_H}}{y + {y_H}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{3} + {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{i \\, {y_H} \\log\\left(\\frac{-i \\, {y_H}^{2} + \\sqrt{{y_H}^{2} + 1} y - i}{i \\, {y_H}^{2} + \\sqrt{{y_H}^{2} + 1} y + i}\\right) + \\sqrt{{y_H}^{2} + 1} \\log\\left(\\frac{y - {y_H}}{y + {y_H}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{3} + {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$$" ], "text/plain": [ "1/2*(I*yH*log((-I*yH^2 + sqrt(yH^2 + 1)*y - I)/(I*yH^2 + sqrt(yH^2 + 1)*y + I)) + sqrt(yH^2 + 1)*log((y - yH)/(y + yH)))/((2*yH^3 + yH)*sqrt(yH^2 + 1))" ] }, "execution_count": 116, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.subs({m: m_yH}).canonicalize_radical().simplify_log()\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Again, we use the identity\n", "$$\n", "i \\ln\\left( \\frac{x - i}{x + i} \\right) = - 2 \\, \\mathrm{arctan}(x) + \\pi\n", "$$\n", "to rewrite $s_2$ as" ] }, { "cell_type": "code", "execution_count": 117, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {y_H} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) - \\sqrt{{y_H}^{2} + 1} \\log\\left(\\frac{y - {y_H}}{y + {y_H}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{3} + {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {y_H} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) - \\sqrt{{y_H}^{2} + 1} \\log\\left(\\frac{y - {y_H}}{y + {y_H}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{3} + {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$$" ], "text/plain": [ "-1/2*(2*yH*arctan(y/sqrt(yH^2 + 1)) - sqrt(yH^2 + 1)*log((y - yH)/(y + yH)))/((2*yH^3 + yH)*sqrt(yH^2 + 1))" ] }, "execution_count": 117, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.subs({I*yH*log((-I*yH^2 + sqrt(yH^2 + 1)*y - I)/(I*yH^2 + sqrt(yH^2 + 1)*y + I)):\n", " -2*yH*atan(y/sqrt(yH^2 + 1))})\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us also replace $\\ln\\left(\\frac{y - y_H}{y + y_H}\\right)$ by $-\\ln\\left(\\frac{y + y_H}{y - y_H}\\right)$\n", "in order to have the same log term as in $s_1(y)$:" ] }, { "cell_type": "code", "execution_count": 118, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {y_H} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) + \\sqrt{{y_H}^{2} + 1} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{3} + {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {y_H} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) + \\sqrt{{y_H}^{2} + 1} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{3} + {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$$" ], "text/plain": [ "-1/2*(2*yH*arctan(y/sqrt(yH^2 + 1)) + sqrt(yH^2 + 1)*log((y + yH)/(y - yH)))/((2*yH^3 + yH)*sqrt(yH^2 + 1))" ] }, "execution_count": 118, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.subs({log((y - yH)/(y + yH)): - log((y + yH)/(y - yH))})\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us check that we have indeed a primitive of $y\\mapsto \\frac{1}{y^4 + y^2 - 2m}$:" ] }, { "cell_type": "code", "execution_count": 119, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{y^{4} - {y_H}^{4} + y^{2} - {y_H}^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{y^{4} - {y_H}^{4} + y^{2} - {y_H}^{2}}$$" ], "text/plain": [ "1/(y^4 - yH^4 + y^2 - yH^2)" ] }, "execution_count": 119, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Ds2 = diff(s2, y).simplify_full()\n", "Ds2" ] }, { "cell_type": "code", "execution_count": 120, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "1/(y^4 + y^2 - 2*m)" ] }, "execution_count": 120, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Ds2.subs({yH: yH_m}).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The full integral is thus" ] }, { "cell_type": "code", "execution_count": 121, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + {\\left(a^{2} - b^{2}\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} \\sqrt{{y_H}^{2} + 1} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right) - 2 \\, {\\left({\\left(a^{2} - b^{2}\\right)} {y_H}^{3} - {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} + b^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} {y_H}\\right)} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{3} + {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + {\\left(a^{2} - b^{2}\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} \\sqrt{{y_H}^{2} + 1} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right) - 2 \\, {\\left({\\left(a^{2} - b^{2}\\right)} {y_H}^{3} - {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} + b^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} {y_H}\\right)} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right)}{2 \\, {\\left(2 \\, {y_H}^{3} + {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$$" ], "text/plain": [ "1/2*((a^2*pf^2 - b^2*qf^2 + (a^2 - b^2)*yH^2 + 2*(a^2 - b^2)*m)*sqrt(yH^2 + 1)*log((y + yH)/(y - yH)) - 2*((a^2 - b^2)*yH^3 - (a^2*pf^2 - b^2*qf^2 - a^2 + b^2 + 2*(a^2 - b^2)*m)*yH)*arctan(y/sqrt(yH^2 + 1)))/((2*yH^3 + yH)*sqrt(yH^2 + 1))" ] }, "execution_count": 121, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Integ0 = (F1*s1 + F2*s2).simplify_full()\n", "Integ0" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "so that the solution is" ] }, { "cell_type": "code", "execution_count": 122, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left({\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{3} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} \\sin\\left({\\Theta_2}\\right) + b^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} {y_H}\\right)} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) + {\\left(4 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H}^{3} + 2 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)\\right)} \\sqrt{{y_H}^{2} + 1}}{4 \\, {\\left(2 \\, {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}^{3} + {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left({\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{3} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} \\sin\\left({\\Theta_2}\\right) + b^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} {y_H}\\right)} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) + {\\left(4 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H}^{3} + 2 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)\\right)} \\sqrt{{y_H}^{2} + 1}}{4 \\, {\\left(2 \\, {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}^{3} + {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$$" ], "text/plain": [ "1/4*(2*((a^2*sin(Th2) - b^2*sin(Th2))*yH^3 - (a^2*pf^2*sin(Th2) - b^2*qf^2*sin(Th2) - a^2*sin(Th2) + b^2*sin(Th2) + 2*(a^2*sin(Th2) - b^2*sin(Th2))*m)*yH)*arctan(y/sqrt(yH^2 + 1)) + (4*(2*C_1*a^2 + 4*C_1*a*b + 2*C_1*b^2 - (a^2*sin(Th2) - b^2*sin(Th2))*y)*yH^3 + 2*(2*C_1*a^2 + 4*C_1*a*b + 2*C_1*b^2 - (a^2*sin(Th2) - b^2*sin(Th2))*y)*yH - (a^2*pf^2*sin(Th2) - b^2*qf^2*sin(Th2) + (a^2*sin(Th2) - b^2*sin(Th2))*yH^2 + 2*(a^2*sin(Th2) - b^2*sin(Th2))*m)*log((y + yH)/(y - yH)))*sqrt(yH^2 + 1))/((2*((a^2 + 2*a*b + b^2)*y^4 + (a^2 + 2*a*b + b^2)*y^2 - 2*(a^2 + 2*a*b + b^2)*m)*yH^3 + ((a^2 + 2*a*b + b^2)*y^4 + (a^2 + 2*a*b + b^2)*y^2 - 2*(a^2 + 2*a*b + b^2)*m)*yH)*sqrt(yH^2 + 1))" ] }, "execution_count": 122, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y_sol(y) = Y_sol0(y).subs({Integ(y): Integ0}).simplify_full()\n", "Y_sol(y)" ] }, { "cell_type": "code", "execution_count": 123, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left({\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{3} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} \\sin\\left({\\Theta_2}\\right) + b^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} {y_H}\\right)} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) + {\\left(4 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H}^{3} + 2 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)\\right)} \\sqrt{{y_H}^{2} + 1}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left({\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{3} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} \\sin\\left({\\Theta_2}\\right) + b^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} {y_H}\\right)} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) + {\\left(4 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H}^{3} + 2 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)\\right)} \\sqrt{{y_H}^{2} + 1}$$" ], "text/plain": [ "2*((a^2*sin(Th2) - b^2*sin(Th2))*yH^3 - (a^2*pf^2*sin(Th2) - b^2*qf^2*sin(Th2) - a^2*sin(Th2) + b^2*sin(Th2) + 2*(a^2*sin(Th2) - b^2*sin(Th2))*m)*yH)*arctan(y/sqrt(yH^2 + 1)) + (4*(2*C_1*a^2 + 4*C_1*a*b + 2*C_1*b^2 - (a^2*sin(Th2) - b^2*sin(Th2))*y)*yH^3 + 2*(2*C_1*a^2 + 4*C_1*a*b + 2*C_1*b^2 - (a^2*sin(Th2) - b^2*sin(Th2))*y)*yH - (a^2*pf^2*sin(Th2) - b^2*qf^2*sin(Th2) + (a^2*sin(Th2) - b^2*sin(Th2))*yH^2 + 2*(a^2*sin(Th2) - b^2*sin(Th2))*m)*log((y + yH)/(y - yH)))*sqrt(yH^2 + 1)" ] }, "execution_count": 123, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y_sol(y).numerator().simplify_full()" ] }, { "cell_type": "code", "execution_count": 124, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1} {\\left(a + b\\right)}^{2} {y_H}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1} {\\left(a + b\\right)}^{2} {y_H}$$" ], "text/plain": [ "4*(y^4 + y^2 - 2*m)*(2*yH^2 + 1)*sqrt(yH^2 + 1)*(a + b)^2*yH" ] }, "execution_count": 124, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y_sol(y).denominator().factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us check that `Y_sol` is indeed a solution of the differential equation for $\\Upsilon$:" ] }, { "cell_type": "code", "execution_count": 125, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$$" ], "text/plain": [ "0 == 0" ] }, "execution_count": 125, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_Y.substitute_function(Y, Y_sol).subs({yH: yH_m}).simplify_full()" ] }, { "cell_type": "code", "execution_count": 126, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1/4*(2*((a^2*sin(Th2) - b^2*sin(Th2))*yH^3 - (a^2*pf^2*sin(Th2) - b^2*qf^2*sin(Th2) - a^2*sin(Th2) + b^2*sin(Th2) + 2*(a^2*sin(Th2) - b^2*sin(Th2))*m)*yH)*arctan(y/sqrt(yH^2 + 1)) + (4*(2*C_1*a^2 + 4*C_1*a*b + 2*C_1*b^2 - (a^2*sin(Th2) - b^2*sin(Th2))*y)*yH^3 + 2*(2*C_1*a^2 + 4*C_1*a*b + 2*C_1*b^2 - (a^2*sin(Th2) - b^2*sin(Th2))*y)*yH - (a^2*pf^2*sin(Th2) - b^2*qf^2*sin(Th2) + (a^2*sin(Th2) - b^2*sin(Th2))*yH^2 + 2*(a^2*sin(Th2) - b^2*sin(Th2))*m)*log((y + yH)/(y - yH)))*sqrt(yH^2 + 1))/((2*((a^2 + 2*a*b + b^2)*y^4 + (a^2 + 2*a*b + b^2)*y^2 - 2*(a^2 + 2*a*b + b^2)*m)*yH^3 + ((a^2 + 2*a*b + b^2)*y^4 + (a^2 + 2*a*b + b^2)*y^2 - 2*(a^2 + 2*a*b + b^2)*m)*yH)*sqrt(yH^2 + 1))\n" ] } ], "source": [ "print(Y_sol(y))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (4.31) (expression of $\\Theta'_1 = \\Upsilon$)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The term involving the constant $C_1$ agrees with that of Eq. (4.31):" ] }, { "cell_type": "code", "execution_count": 127, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "1/(y^4 + y^2 - 2*m)" ] }, "execution_count": 127, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = Y_sol(y).coefficient(C_1).simplify_full()\n", "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us remove it from $\\Upsilon$ and divide the result by $\\sin(2\\Theta_0)$:" ] }, { "cell_type": "code", "execution_count": 128, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left({\\left(a^{2} - b^{2}\\right)} {y_H}^{3} - {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} + b^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} {y_H}\\right)} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) - {\\left(4 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H}^{3} + 2 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H} + {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + {\\left(a^{2} - b^{2}\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)\\right)} \\sqrt{{y_H}^{2} + 1}}{4 \\, {\\left(2 \\, {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}^{3} + {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left({\\left(a^{2} - b^{2}\\right)} {y_H}^{3} - {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} + b^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} {y_H}\\right)} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) - {\\left(4 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H}^{3} + 2 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H} + {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + {\\left(a^{2} - b^{2}\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)\\right)} \\sqrt{{y_H}^{2} + 1}}{4 \\, {\\left(2 \\, {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}^{3} + {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$$" ], "text/plain": [ "1/4*(2*((a^2 - b^2)*yH^3 - (a^2*pf^2 - b^2*qf^2 - a^2 + b^2 + 2*(a^2 - b^2)*m)*yH)*arctan(y/sqrt(yH^2 + 1)) - (4*(a^2 - b^2)*y*yH^3 + 2*(a^2 - b^2)*y*yH + (a^2*pf^2 - b^2*qf^2 + (a^2 - b^2)*yH^2 + 2*(a^2 - b^2)*m)*log((y + yH)/(y - yH)))*sqrt(yH^2 + 1))/((2*((a^2 + 2*a*b + b^2)*y^4 + (a^2 + 2*a*b + b^2)*y^2 - 2*(a^2 + 2*a*b + b^2)*m)*yH^3 + ((a^2 + 2*a*b + b^2)*y^4 + (a^2 + 2*a*b + b^2)*y^2 - 2*(a^2 + 2*a*b + b^2)*m)*yH)*sqrt(yH^2 + 1))" ] }, "execution_count": 128, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y1 = ((Y_sol(y) - s*C_1)/sin(Th2)).simplify_full()\n", "Y1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coefficient of the arctan term is" ] }, { "cell_type": "code", "execution_count": 129, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} {y_H}^{2} + b^{2} {y_H}^{2} + 2 \\, a^{2} m - 2 \\, b^{2} m - a^{2} + b^{2}}{2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1} {\\left(a + b\\right)}^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} {y_H}^{2} + b^{2} {y_H}^{2} + 2 \\, a^{2} m - 2 \\, b^{2} m - a^{2} + b^{2}}{2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1} {\\left(a + b\\right)}^{2}}$$" ], "text/plain": [ "-1/2*(a^2*pf^2 - b^2*qf^2 - a^2*yH^2 + b^2*yH^2 + 2*a^2*m - 2*b^2*m - a^2 + b^2)/((y^4 + y^2 - 2*m)*(2*yH^2 + 1)*sqrt(yH^2 + 1)*(a + b)^2)" ] }, "execution_count": 129, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = Y1.coefficient(arctan(y/sqrt(yH^2 + 1))).simplify_full().factor()\n", "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The numerator of this term agrees with Eq. (4.31), once we express $m$ in terms of $y_H$:" ] }, { "cell_type": "code", "execution_count": 130, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left(a^{2} - b^{2}\\right)} {y_H}^{4} - a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} + a^{2} - b^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left(a^{2} - b^{2}\\right)} {y_H}^{4} - a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} + a^{2} - b^{2}$$" ], "text/plain": [ "-(a^2 - b^2)*yH^4 - a^2*pf^2 + b^2*qf^2 + a^2 - b^2" ] }, "execution_count": 130, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s.numerator().subs({m: m_yH}).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The denominator agrees with Eq. (4.31) as well:" ] }, { "cell_type": "code", "execution_count": 131, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1} {\\left(a + b\\right)}^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1} {\\left(a + b\\right)}^{2}$$" ], "text/plain": [ "2*(y^4 + y^2 - 2*m)*(2*yH^2 + 1)*sqrt(yH^2 + 1)*(a + b)^2" ] }, "execution_count": 131, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s.denominator()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us remove the arctan term from $\\Upsilon$:" ] }, { "cell_type": "code", "execution_count": 132, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{4 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H}^{3} + 2 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H} + {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + {\\left(a^{2} - b^{2}\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)}{4 \\, {\\left(2 \\, {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}^{3} + {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{4 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H}^{3} + 2 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H} + {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + {\\left(a^{2} - b^{2}\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)}{4 \\, {\\left(2 \\, {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}^{3} + {\\left({\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{4} + {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} y^{2} - 2 \\, {\\left(a^{2} + 2 \\, a b + b^{2}\\right)} m\\right)} {y_H}\\right)}}$$" ], "text/plain": [ "-1/4*(4*(a^2 - b^2)*y*yH^3 + 2*(a^2 - b^2)*y*yH + (a^2*pf^2 - b^2*qf^2 + (a^2 - b^2)*yH^2 + 2*(a^2 - b^2)*m)*log((y + yH)/(y - yH)))/(2*((a^2 + 2*a*b + b^2)*y^4 + (a^2 + 2*a*b + b^2)*y^2 - 2*(a^2 + 2*a*b + b^2)*m)*yH^3 + ((a^2 + 2*a*b + b^2)*y^4 + (a^2 + 2*a*b + b^2)*y^2 - 2*(a^2 + 2*a*b + b^2)*m)*yH)" ] }, "execution_count": 132, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y2 = (Y1 - s*arctan(y/sqrt(yH^2 + 1))).simplify_full()\n", "Y2" ] }, { "cell_type": "code", "execution_count": 133, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H}^{3} - 2 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H} - {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + {\\left(a^{2} - b^{2}\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H}^{3} - 2 \\, {\\left(a^{2} - b^{2}\\right)} y {y_H} - {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + {\\left(a^{2} - b^{2}\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} - b^{2}\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)$$" ], "text/plain": [ "-4*(a^2 - b^2)*y*yH^3 - 2*(a^2 - b^2)*y*yH - (a^2*pf^2 - b^2*qf^2 + (a^2 - b^2)*yH^2 + 2*(a^2 - b^2)*m)*log((y + yH)/(y - yH))" ] }, "execution_count": 133, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y2.numerator().simplify_full()" ] }, { "cell_type": "code", "execution_count": 134, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} {\\left(a + b\\right)}^{2} {y_H}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} {\\left(a + b\\right)}^{2} {y_H}$$" ], "text/plain": [ "4*(y^4 + y^2 - 2*m)*(2*yH^2 + 1)*(a + b)^2*yH" ] }, "execution_count": 134, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y2.denominator().factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coefficient of the log term is" ] }, { "cell_type": "code", "execution_count": 135, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + a^{2} {y_H}^{2} - b^{2} {y_H}^{2} + 2 \\, a^{2} m - 2 \\, b^{2} m}{4 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} {\\left(a + b\\right)}^{2} {y_H}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} + a^{2} {y_H}^{2} - b^{2} {y_H}^{2} + 2 \\, a^{2} m - 2 \\, b^{2} m}{4 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} {\\left(a + b\\right)}^{2} {y_H}}$$" ], "text/plain": [ "-1/4*(a^2*pf^2 - b^2*qf^2 + a^2*yH^2 - b^2*yH^2 + 2*a^2*m - 2*b^2*m)/((y^4 + y^2 - 2*m)*(2*yH^2 + 1)*(a + b)^2*yH)" ] }, "execution_count": 135, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = Y2.coefficient(log((y + yH)/(y - yH))).simplify_full().factor()\n", "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The numerator and denominator both agree with Eq. (4.31):" ] }, { "cell_type": "code", "execution_count": 136, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left(a^{2} - b^{2}\\right)} {y_H}^{4} - a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} - 2 \\, {\\left(a^{2} - b^{2}\\right)} {y_H}^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left(a^{2} - b^{2}\\right)} {y_H}^{4} - a^{2} {\\mathfrak{p}}^{2} + b^{2} {\\mathfrak{q}}^{2} - 2 \\, {\\left(a^{2} - b^{2}\\right)} {y_H}^{2}$$" ], "text/plain": [ "-(a^2 - b^2)*yH^4 - a^2*pf^2 + b^2*qf^2 - 2*(a^2 - b^2)*yH^2" ] }, "execution_count": 136, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s.numerator().subs({m: m_yH}).simplify_full()" ] }, { "cell_type": "code", "execution_count": 137, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} {\\left(a + b\\right)}^{2} {y_H}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(2 \\, {y_H}^{2} + 1\\right)} {\\left(a + b\\right)}^{2} {y_H}$$" ], "text/plain": [ "4*(y^4 + y^2 - 2*m)*(2*yH^2 + 1)*(a + b)^2*yH" ] }, "execution_count": 137, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s.denominator()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Given that \n", "$$ \\mathrm{artanh}\\, x = \\frac{1}{2} \\ln\\left( \\frac{1 + x}{1 - x} \\right) $$\n", "we have\n", "$$\n", " \\ln \\left( \\frac{x + 1}{x - 1} \\right) = 2\\, \\mathrm{artanh}\\left(\\frac{1}{x}\\right)\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Hence the term in $\\ln\\left(\\frac{y + y_H}{y - y_H}\\right)$ agrees with the corresponding term in Eq. (4.30)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Finally, the last term in $\\Upsilon$ is" ] }, { "cell_type": "code", "execution_count": 138, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(a - b\\right)} y}{2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(a + b\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(a - b\\right)} y}{2 \\, {\\left(y^{4} + y^{2} - 2 \\, m\\right)} {\\left(a + b\\right)}}$$" ], "text/plain": [ "-1/2*(a - b)*y/((y^4 + y^2 - 2*m)*(a + b))" ] }, "execution_count": 138, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Y3 = (Y2 - s*log((y + yH)/(y - yH))).simplify_full()\n", "Y3.factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This term agrees with Eq. (4.31), given the simplification\n", "$\\frac{a^2 - b^2}{(a + b)^2} = \\frac{a - b}{a + b}$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Conjugate momenta" ] }, { "cell_type": "code", "execution_count": 139, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def conjugate_momenta(lagr, qs, var):\n", " r\"\"\"\n", " Compute the conjugate momenta from a given Lagrangian.\n", "\n", " INPUT:\n", "\n", " - ``lagr`` -- symbolic expression representing the Lagrangian density\n", " - ``qs`` -- either a single symbolic function or a list/tuple of\n", " symbolic functions, representing the `q`'s; these functions must\n", " appear in ``lagr`` up to at most their first derivatives\n", " - ``var`` -- either a single variable, typically `t` (1-dimensional\n", " problem) or a list/tuple of symbolic variables; in the latter case the\n", " time coordinate must the first one\n", "\n", " OUTPUT:\n", "\n", " - list of conjugate momenta; if only one function is involved, the\n", " single conjugate momentum is returned instead.\n", "\n", " \"\"\"\n", " if not isinstance(qs, (list, tuple)):\n", " qs = [qs]\n", " if not isinstance(var, (list, tuple)):\n", " var = [var]\n", " n = len(qs)\n", " d = len(var)\n", " dqvt = [SR.var('qxxxx{}_t'.format(q)) for q in qs]\n", " subs = {diff(qs[i](*var), var[0]): dqvt[i] for i in range(n)}\n", " subs_inv = {dqvt[i]: diff(qs[i](*var), var[0]) for i in range(n)}\n", " lg = lagr.substitute(subs)\n", " ps = [diff(lg, dotq).simplify_full().substitute(subs_inv) for dotq in dqvt]\n", " if n == 1:\n", " return ps[0]\n", " return ps" ] }, { "cell_type": "code", "execution_count": 140, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-{\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) - {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + 2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right), b^{2} {\\chi_0}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) + b^{2} {\\chi_0}^{2} y^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - 2 \\, b^{2} {\\chi_0}^{2} m \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)\\right]$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-{\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) - {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + 2 \\, {\\left(a^{2} {\\chi_0}^{2} - a^{2}\\right)} m \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right), b^{2} {\\chi_0}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) + b^{2} {\\chi_0}^{2} y^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - 2 \\, b^{2} {\\chi_0}^{2} m \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)\\right]$$" ], "text/plain": [ "[-(a^2*ch0^2 - a^2)*y^4*diff(Phi_1(y), y) - (a^2*ch0^2 - a^2)*y^2*diff(Phi_1(y), y) + 2*(a^2*ch0^2 - a^2)*m*diff(Phi_1(y), y),\n", " b^2*ch0^2*y^4*diff(Psi_1(y), y) + b^2*ch0^2*y^2*diff(Psi_1(y), y) - 2*b^2*ch0^2*m*diff(Psi_1(y), y)]" ] }, "execution_count": 140, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pis = conjugate_momenta(L_2, [Phi_1, Psi_1], y)\n", "pis" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "$\\pi^y_\\Phi$:" ] }, { "cell_type": "code", "execution_count": 141, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left(a {\\chi_0}^{2} - a\\right)} {\\mathfrak{p}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left(a {\\chi_0}^{2} - a\\right)} {\\mathfrak{p}}$$" ], "text/plain": [ "-(a*ch0^2 - a)*pf" ] }, "execution_count": 141, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_Phi_y = (pis[0]/a).substitute_function(Phi_1, Phi1_sol).simplify_full()\n", "pi_Phi_y" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "$\\pi_\\Psi^y$:" ] }, { "cell_type": "code", "execution_count": 142, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}b {\\chi_0}^{2} {\\mathfrak{q}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}b {\\chi_0}^{2} {\\mathfrak{q}}$$" ], "text/plain": [ "b*ch0^2*qf" ] }, "execution_count": 142, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_Psi_y = (pis[1]/b).substitute_function(Psi_1, Psi1_sol).simplify_full()\n", "pi_Psi_y" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (4.33)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We start from $\\pi^y_\\Theta$ as given by Eq. (4.32):" ] }, { "cell_type": "code", "execution_count": 143, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left({\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{3} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} \\sin\\left({\\Theta_2}\\right) + b^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} {y_H}\\right)} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) + {\\left(4 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H}^{3} + 2 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)\\right)} \\sqrt{{y_H}^{2} + 1}}{4 \\, {\\left(2 \\, {y_H}^{3} + {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left({\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{3} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) - a^{2} \\sin\\left({\\Theta_2}\\right) + b^{2} \\sin\\left({\\Theta_2}\\right) + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} {y_H}\\right)} \\arctan\\left(\\frac{y}{\\sqrt{{y_H}^{2} + 1}}\\right) + {\\left(4 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H}^{3} + 2 \\, {\\left(2 \\, C_{1} a^{2} + 4 \\, C_{1} a b + 2 \\, C_{1} b^{2} - {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y\\right)} {y_H} - {\\left(a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) + {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{2} + 2 \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m\\right)} \\log\\left(\\frac{y + {y_H}}{y - {y_H}}\\right)\\right)} \\sqrt{{y_H}^{2} + 1}}{4 \\, {\\left(2 \\, {y_H}^{3} + {y_H}\\right)} \\sqrt{{y_H}^{2} + 1}}$$" ], "text/plain": [ "1/4*(2*((a^2*sin(Th2) - b^2*sin(Th2))*yH^3 - (a^2*pf^2*sin(Th2) - b^2*qf^2*sin(Th2) - a^2*sin(Th2) + b^2*sin(Th2) + 2*(a^2*sin(Th2) - b^2*sin(Th2))*m)*yH)*arctan(y/sqrt(yH^2 + 1)) + (4*(2*C_1*a^2 + 4*C_1*a*b + 2*C_1*b^2 - (a^2*sin(Th2) - b^2*sin(Th2))*y)*yH^3 + 2*(2*C_1*a^2 + 4*C_1*a*b + 2*C_1*b^2 - (a^2*sin(Th2) - b^2*sin(Th2))*y)*yH - (a^2*pf^2*sin(Th2) - b^2*qf^2*sin(Th2) + (a^2*sin(Th2) - b^2*sin(Th2))*yH^2 + 2*(a^2*sin(Th2) - b^2*sin(Th2))*m)*log((y + yH)/(y - yH)))*sqrt(yH^2 + 1))/((2*yH^3 + yH)*sqrt(yH^2 + 1))" ] }, "execution_count": 143, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_Theta = ((y^4 + y^2 - 2*m)*(a + b)^2*Y_sol(y)).simplify_full()\n", "pi_Theta " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us perform an expansion in $1/y$ for $y\\rightarrow +\\infty$:" ] }, { "cell_type": "code", "execution_count": 144, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y - \\frac{a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)}{2 \\, y} - \\frac{\\pi a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - \\pi b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) - \\pi a^{2} \\sin\\left({\\Theta_2}\\right) + \\pi b^{2} \\sin\\left({\\Theta_2}\\right) - {\\left(\\pi a^{2} \\sin\\left({\\Theta_2}\\right) - \\pi b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{2} + 2 \\, {\\left(\\pi a^{2} \\sin\\left({\\Theta_2}\\right) - \\pi b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m - 4 \\, {\\left(C_{1} a^{2} + 2 \\, C_{1} a b + C_{1} b^{2} + 2 \\, {\\left(C_{1} a^{2} + 2 \\, C_{1} a b + C_{1} b^{2}\\right)} {y_H}^{2}\\right)} \\sqrt{{y_H}^{2} + 1}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, {\\left(a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} y - \\frac{a^{2} \\sin\\left({\\Theta_2}\\right) - b^{2} \\sin\\left({\\Theta_2}\\right)}{2 \\, y} - \\frac{\\pi a^{2} {\\mathfrak{p}}^{2} \\sin\\left({\\Theta_2}\\right) - \\pi b^{2} {\\mathfrak{q}}^{2} \\sin\\left({\\Theta_2}\\right) - \\pi a^{2} \\sin\\left({\\Theta_2}\\right) + \\pi b^{2} \\sin\\left({\\Theta_2}\\right) - {\\left(\\pi a^{2} \\sin\\left({\\Theta_2}\\right) - \\pi b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} {y_H}^{2} + 2 \\, {\\left(\\pi a^{2} \\sin\\left({\\Theta_2}\\right) - \\pi b^{2} \\sin\\left({\\Theta_2}\\right)\\right)} m - 4 \\, {\\left(C_{1} a^{2} + 2 \\, C_{1} a b + C_{1} b^{2} + 2 \\, {\\left(C_{1} a^{2} + 2 \\, C_{1} a b + C_{1} b^{2}\\right)} {y_H}^{2}\\right)} \\sqrt{{y_H}^{2} + 1}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}}$$" ], "text/plain": [ "-1/2*(a^2*sin(Th2) - b^2*sin(Th2))*y - 1/2*(a^2*sin(Th2) - b^2*sin(Th2))/y - 1/4*(pi*a^2*pf^2*sin(Th2) - pi*b^2*qf^2*sin(Th2) - pi*a^2*sin(Th2) + pi*b^2*sin(Th2) - (pi*a^2*sin(Th2) - pi*b^2*sin(Th2))*yH^2 + 2*(pi*a^2*sin(Th2) - pi*b^2*sin(Th2))*m - 4*(C_1*a^2 + 2*C_1*a*b + C_1*b^2 + 2*(C_1*a^2 + 2*C_1*a*b + C_1*b^2)*yH^2)*sqrt(yH^2 + 1))/((2*yH^2 + 1)*sqrt(yH^2 + 1))" ] }, "execution_count": 144, "metadata": { }, "output_type": "execute_result" } ], "source": [ "u = var('u')\n", "assume(u > 0)\n", "s = pi_Theta.subs({y: 1/u}).simplify_log()\n", "s = s.taylor(u, 0, 2)\n", "s = s.subs({u: 1/y})\n", "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We consider $\\frac{\\pi^y_\\Theta}{\\sin(2\\Theta_0)}$:" ] }, { "cell_type": "code", "execution_count": 145, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\pi a^{2} {\\mathfrak{p}}^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} + \\frac{\\pi b^{2} {\\mathfrak{q}}^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} + \\frac{\\pi a^{2} {y_H}^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} - \\frac{\\pi b^{2} {y_H}^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} - \\frac{1}{2} \\, a^{2} y + \\frac{1}{2} \\, b^{2} y + \\frac{2 \\, C_{1} a^{2} {y_H}^{2}}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{4 \\, C_{1} a b {y_H}^{2}}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{2 \\, C_{1} b^{2} {y_H}^{2}}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} - \\frac{\\pi a^{2} m}{2 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} + \\frac{\\pi b^{2} m}{2 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} + \\frac{\\pi a^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} - \\frac{\\pi b^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} - \\frac{a^{2}}{2 \\, y} + \\frac{b^{2}}{2 \\, y} + \\frac{C_{1} a^{2}}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{2 \\, C_{1} a b}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{C_{1} b^{2}}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\pi a^{2} {\\mathfrak{p}}^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} + \\frac{\\pi b^{2} {\\mathfrak{q}}^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} + \\frac{\\pi a^{2} {y_H}^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} - \\frac{\\pi b^{2} {y_H}^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} - \\frac{1}{2} \\, a^{2} y + \\frac{1}{2} \\, b^{2} y + \\frac{2 \\, C_{1} a^{2} {y_H}^{2}}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{4 \\, C_{1} a b {y_H}^{2}}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{2 \\, C_{1} b^{2} {y_H}^{2}}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} - \\frac{\\pi a^{2} m}{2 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} + \\frac{\\pi b^{2} m}{2 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} + \\frac{\\pi a^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} - \\frac{\\pi b^{2}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}} - \\frac{a^{2}}{2 \\, y} + \\frac{b^{2}}{2 \\, y} + \\frac{C_{1} a^{2}}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{2 \\, C_{1} a b}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)} + \\frac{C_{1} b^{2}}{{\\left(2 \\, {y_H}^{2} + 1\\right)} \\sin\\left({\\Theta_2}\\right)}$$" ], "text/plain": [ "-1/4*pi*a^2*pf^2/((2*yH^2 + 1)*sqrt(yH^2 + 1)) + 1/4*pi*b^2*qf^2/((2*yH^2 + 1)*sqrt(yH^2 + 1)) + 1/4*pi*a^2*yH^2/((2*yH^2 + 1)*sqrt(yH^2 + 1)) - 1/4*pi*b^2*yH^2/((2*yH^2 + 1)*sqrt(yH^2 + 1)) - 1/2*a^2*y + 1/2*b^2*y + 2*C_1*a^2*yH^2/((2*yH^2 + 1)*sin(Th2)) + 4*C_1*a*b*yH^2/((2*yH^2 + 1)*sin(Th2)) + 2*C_1*b^2*yH^2/((2*yH^2 + 1)*sin(Th2)) - 1/2*pi*a^2*m/((2*yH^2 + 1)*sqrt(yH^2 + 1)) + 1/2*pi*b^2*m/((2*yH^2 + 1)*sqrt(yH^2 + 1)) + 1/4*pi*a^2/((2*yH^2 + 1)*sqrt(yH^2 + 1)) - 1/4*pi*b^2/((2*yH^2 + 1)*sqrt(yH^2 + 1)) - 1/2*a^2/y + 1/2*b^2/y + C_1*a^2/((2*yH^2 + 1)*sin(Th2)) + 2*C_1*a*b/((2*yH^2 + 1)*sin(Th2)) + C_1*b^2/((2*yH^2 + 1)*sin(Th2))" ] }, "execution_count": 145, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = (s/sin(Th2)).expand()\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The term in factor of $C_1$ is" ] }, { "cell_type": "code", "execution_count": 146, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(a + b\\right)}^{2}}{\\sin\\left({\\Theta_2}\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(a + b\\right)}^{2}}{\\sin\\left({\\Theta_2}\\right)}$$" ], "text/plain": [ "(a + b)^2/sin(Th2)" ] }, "execution_count": 146, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1.coefficient(C_1).factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Hence this terms agrees with Eq. (4.32).\n", "We remove it from the main term:" ] }, { "cell_type": "code", "execution_count": 147, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{4 \\, {\\left({\\left(a^{2} - b^{2}\\right)} y^{2} + a^{2} - b^{2}\\right)} {y_H}^{4} + 2 \\, {\\left(a^{2} - b^{2}\\right)} y^{2} + 6 \\, {\\left({\\left(a^{2} - b^{2}\\right)} y^{2} + a^{2} - b^{2}\\right)} {y_H}^{2} + 2 \\, a^{2} - 2 \\, b^{2} - {\\left({\\left(\\pi a^{2} - \\pi b^{2}\\right)} y {y_H}^{2} - {\\left(\\pi a^{2} {\\mathfrak{p}}^{2} - \\pi b^{2} {\\mathfrak{q}}^{2} - \\pi a^{2} + \\pi b^{2} + 2 \\, {\\left(\\pi a^{2} - \\pi b^{2}\\right)} m\\right)} y\\right)} \\sqrt{{y_H}^{2} + 1}}{4 \\, {\\left(2 \\, y {y_H}^{4} + 3 \\, y {y_H}^{2} + y\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{4 \\, {\\left({\\left(a^{2} - b^{2}\\right)} y^{2} + a^{2} - b^{2}\\right)} {y_H}^{4} + 2 \\, {\\left(a^{2} - b^{2}\\right)} y^{2} + 6 \\, {\\left({\\left(a^{2} - b^{2}\\right)} y^{2} + a^{2} - b^{2}\\right)} {y_H}^{2} + 2 \\, a^{2} - 2 \\, b^{2} - {\\left({\\left(\\pi a^{2} - \\pi b^{2}\\right)} y {y_H}^{2} - {\\left(\\pi a^{2} {\\mathfrak{p}}^{2} - \\pi b^{2} {\\mathfrak{q}}^{2} - \\pi a^{2} + \\pi b^{2} + 2 \\, {\\left(\\pi a^{2} - \\pi b^{2}\\right)} m\\right)} y\\right)} \\sqrt{{y_H}^{2} + 1}}{4 \\, {\\left(2 \\, y {y_H}^{4} + 3 \\, y {y_H}^{2} + y\\right)}}$$" ], "text/plain": [ "-1/4*(4*((a^2 - b^2)*y^2 + a^2 - b^2)*yH^4 + 2*(a^2 - b^2)*y^2 + 6*((a^2 - b^2)*y^2 + a^2 - b^2)*yH^2 + 2*a^2 - 2*b^2 - ((pi*a^2 - pi*b^2)*y*yH^2 - (pi*a^2*pf^2 - pi*b^2*qf^2 - pi*a^2 + pi*b^2 + 2*(pi*a^2 - pi*b^2)*m)*y)*sqrt(yH^2 + 1))/(2*y*yH^4 + 3*y*yH^2 + y)" ] }, "execution_count": 147, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = (s1 - s1.coefficient(C_1)*C_1).simplify_full()\n", "s2" ] }, { "cell_type": "code", "execution_count": 148, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4 \\, {\\left({\\left(a^{2} - b^{2}\\right)} y^{2} + a^{2} - b^{2}\\right)} {y_H}^{4} - 2 \\, {\\left(a^{2} - b^{2}\\right)} y^{2} - 6 \\, {\\left({\\left(a^{2} - b^{2}\\right)} y^{2} + a^{2} - b^{2}\\right)} {y_H}^{2} - 2 \\, a^{2} + 2 \\, b^{2} + {\\left({\\left(\\pi a^{2} - \\pi b^{2}\\right)} y {y_H}^{2} - {\\left(\\pi a^{2} {\\mathfrak{p}}^{2} - \\pi b^{2} {\\mathfrak{q}}^{2} - \\pi a^{2} + \\pi b^{2} + 2 \\, {\\left(\\pi a^{2} - \\pi b^{2}\\right)} m\\right)} y\\right)} \\sqrt{{y_H}^{2} + 1}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4 \\, {\\left({\\left(a^{2} - b^{2}\\right)} y^{2} + a^{2} - b^{2}\\right)} {y_H}^{4} - 2 \\, {\\left(a^{2} - b^{2}\\right)} y^{2} - 6 \\, {\\left({\\left(a^{2} - b^{2}\\right)} y^{2} + a^{2} - b^{2}\\right)} {y_H}^{2} - 2 \\, a^{2} + 2 \\, b^{2} + {\\left({\\left(\\pi a^{2} - \\pi b^{2}\\right)} y {y_H}^{2} - {\\left(\\pi a^{2} {\\mathfrak{p}}^{2} - \\pi b^{2} {\\mathfrak{q}}^{2} - \\pi a^{2} + \\pi b^{2} + 2 \\, {\\left(\\pi a^{2} - \\pi b^{2}\\right)} m\\right)} y\\right)} \\sqrt{{y_H}^{2} + 1}$$" ], "text/plain": [ "-4*((a^2 - b^2)*y^2 + a^2 - b^2)*yH^4 - 2*(a^2 - b^2)*y^2 - 6*((a^2 - b^2)*y^2 + a^2 - b^2)*yH^2 - 2*a^2 + 2*b^2 + ((pi*a^2 - pi*b^2)*y*yH^2 - (pi*a^2*pf^2 - pi*b^2*qf^2 - pi*a^2 + pi*b^2 + 2*(pi*a^2 - pi*b^2)*m)*y)*sqrt(yH^2 + 1)" ] }, "execution_count": 148, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2.numerator().simplify_full()" ] }, { "cell_type": "code", "execution_count": 149, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} {\\left({y_H}^{2} + 1\\right)} y$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} {\\left({y_H}^{2} + 1\\right)} y$$" ], "text/plain": [ "4*(2*yH^2 + 1)*(yH^2 + 1)*y" ] }, "execution_count": 149, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2.denominator().factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let divide both the numerator and denominator by $y$" ] }, { "cell_type": "code", "execution_count": 150, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4 \\, a^{2} y {y_H}^{4} + 4 \\, b^{2} y {y_H}^{4} - \\pi \\sqrt{{y_H}^{2} + 1} a^{2} {\\mathfrak{p}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {\\mathfrak{q}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} {y_H}^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {y_H}^{2} - 6 \\, a^{2} y {y_H}^{2} + 6 \\, b^{2} y {y_H}^{2} - \\frac{4 \\, a^{2} {y_H}^{4}}{y} + \\frac{4 \\, b^{2} {y_H}^{4}}{y} - 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} a^{2} m + 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} b^{2} m + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2} - 2 \\, a^{2} y + 2 \\, b^{2} y - \\frac{6 \\, a^{2} {y_H}^{2}}{y} + \\frac{6 \\, b^{2} {y_H}^{2}}{y} - \\frac{2 \\, a^{2}}{y} + \\frac{2 \\, b^{2}}{y}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4 \\, a^{2} y {y_H}^{4} + 4 \\, b^{2} y {y_H}^{4} - \\pi \\sqrt{{y_H}^{2} + 1} a^{2} {\\mathfrak{p}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {\\mathfrak{q}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} {y_H}^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {y_H}^{2} - 6 \\, a^{2} y {y_H}^{2} + 6 \\, b^{2} y {y_H}^{2} - \\frac{4 \\, a^{2} {y_H}^{4}}{y} + \\frac{4 \\, b^{2} {y_H}^{4}}{y} - 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} a^{2} m + 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} b^{2} m + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2} - 2 \\, a^{2} y + 2 \\, b^{2} y - \\frac{6 \\, a^{2} {y_H}^{2}}{y} + \\frac{6 \\, b^{2} {y_H}^{2}}{y} - \\frac{2 \\, a^{2}}{y} + \\frac{2 \\, b^{2}}{y}$$" ], "text/plain": [ "-4*a^2*y*yH^4 + 4*b^2*y*yH^4 - pi*sqrt(yH^2 + 1)*a^2*pf^2 + pi*sqrt(yH^2 + 1)*b^2*qf^2 + pi*sqrt(yH^2 + 1)*a^2*yH^2 - pi*sqrt(yH^2 + 1)*b^2*yH^2 - 6*a^2*y*yH^2 + 6*b^2*y*yH^2 - 4*a^2*yH^4/y + 4*b^2*yH^4/y - 2*pi*sqrt(yH^2 + 1)*a^2*m + 2*pi*sqrt(yH^2 + 1)*b^2*m + pi*sqrt(yH^2 + 1)*a^2 - pi*sqrt(yH^2 + 1)*b^2 - 2*a^2*y + 2*b^2*y - 6*a^2*yH^2/y + 6*b^2*yH^2/y - 2*a^2/y + 2*b^2/y" ] }, "execution_count": 150, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2n = (s2.numerator()/y).expand()\n", "s2n" ] }, { "cell_type": "code", "execution_count": 151, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} {\\left({y_H}^{2} + 1\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} {\\left({y_H}^{2} + 1\\right)}$$" ], "text/plain": [ "4*(2*yH^2 + 1)*(yH^2 + 1)" ] }, "execution_count": 151, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2d = (s2.denominator()/y).factor()\n", "s2d" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coefficient of the term in $y$ is" ] }, { "cell_type": "code", "execution_count": 152, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, {\\left(a + b\\right)} {\\left(a - b\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, {\\left(a + b\\right)} {\\left(a - b\\right)}$$" ], "text/plain": [ "-1/2*(a + b)*(a - b)" ] }, "execution_count": 152, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = s2n.coefficient(y).factor()\n", "s/s2d" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This is in agreement with Eq. (4.33)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We remove it:" ] }, { "cell_type": "code", "execution_count": 153, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\pi \\sqrt{{y_H}^{2} + 1} a^{2} {\\mathfrak{p}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {\\mathfrak{q}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} {y_H}^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {y_H}^{2} - \\frac{4 \\, a^{2} {y_H}^{4}}{y} + \\frac{4 \\, b^{2} {y_H}^{4}}{y} - 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} a^{2} m + 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} b^{2} m + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2} - \\frac{6 \\, a^{2} {y_H}^{2}}{y} + \\frac{6 \\, b^{2} {y_H}^{2}}{y} - \\frac{2 \\, a^{2}}{y} + \\frac{2 \\, b^{2}}{y}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\pi \\sqrt{{y_H}^{2} + 1} a^{2} {\\mathfrak{p}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {\\mathfrak{q}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} {y_H}^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {y_H}^{2} - \\frac{4 \\, a^{2} {y_H}^{4}}{y} + \\frac{4 \\, b^{2} {y_H}^{4}}{y} - 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} a^{2} m + 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} b^{2} m + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2} - \\frac{6 \\, a^{2} {y_H}^{2}}{y} + \\frac{6 \\, b^{2} {y_H}^{2}}{y} - \\frac{2 \\, a^{2}}{y} + \\frac{2 \\, b^{2}}{y}$$" ], "text/plain": [ "-pi*sqrt(yH^2 + 1)*a^2*pf^2 + pi*sqrt(yH^2 + 1)*b^2*qf^2 + pi*sqrt(yH^2 + 1)*a^2*yH^2 - pi*sqrt(yH^2 + 1)*b^2*yH^2 - 4*a^2*yH^4/y + 4*b^2*yH^4/y - 2*pi*sqrt(yH^2 + 1)*a^2*m + 2*pi*sqrt(yH^2 + 1)*b^2*m + pi*sqrt(yH^2 + 1)*a^2 - pi*sqrt(yH^2 + 1)*b^2 - 6*a^2*yH^2/y + 6*b^2*yH^2/y - 2*a^2/y + 2*b^2/y" ] }, "execution_count": 153, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s3n = (s2n - s*y).simplify_full().expand()\n", "s3n" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coefficient of the term in $1/y$ is" ] }, { "cell_type": "code", "execution_count": 154, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, {\\left(a + b\\right)} {\\left(a - b\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, {\\left(a + b\\right)} {\\left(a - b\\right)}$$" ], "text/plain": [ "-1/2*(a + b)*(a - b)" ] }, "execution_count": 154, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = s3n.coefficient(y^(-1)).factor()\n", "s/s2d" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This is in agreement with Eq. (4.33)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Finally the remaining term is" ] }, { "cell_type": "code", "execution_count": 155, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\pi \\sqrt{{y_H}^{2} + 1} a^{2} {\\mathfrak{p}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {\\mathfrak{q}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} {y_H}^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {y_H}^{2} - 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} a^{2} m + 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} b^{2} m + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\pi \\sqrt{{y_H}^{2} + 1} a^{2} {\\mathfrak{p}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {\\mathfrak{q}}^{2} + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} {y_H}^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2} {y_H}^{2} - 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} a^{2} m + 2 \\, \\pi \\sqrt{{y_H}^{2} + 1} b^{2} m + \\pi \\sqrt{{y_H}^{2} + 1} a^{2} - \\pi \\sqrt{{y_H}^{2} + 1} b^{2}$$" ], "text/plain": [ "-pi*sqrt(yH^2 + 1)*a^2*pf^2 + pi*sqrt(yH^2 + 1)*b^2*qf^2 + pi*sqrt(yH^2 + 1)*a^2*yH^2 - pi*sqrt(yH^2 + 1)*b^2*yH^2 - 2*pi*sqrt(yH^2 + 1)*a^2*m + 2*pi*sqrt(yH^2 + 1)*b^2*m + pi*sqrt(yH^2 + 1)*a^2 - pi*sqrt(yH^2 + 1)*b^2" ] }, "execution_count": 155, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s4n = (s3n - s/y).simplify_full().expand()\n", "s4n" ] }, { "cell_type": "code", "execution_count": 156, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\pi {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} {y_H}^{2} + b^{2} {y_H}^{2} + 2 \\, a^{2} m - 2 \\, b^{2} m - a^{2} + b^{2}\\right)} \\sqrt{{y_H}^{2} + 1}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\pi {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} {y_H}^{2} + b^{2} {y_H}^{2} + 2 \\, a^{2} m - 2 \\, b^{2} m - a^{2} + b^{2}\\right)} \\sqrt{{y_H}^{2} + 1}$$" ], "text/plain": [ "-pi*(a^2*pf^2 - b^2*qf^2 - a^2*yH^2 + b^2*yH^2 + 2*a^2*m - 2*b^2*m - a^2 + b^2)*sqrt(yH^2 + 1)" ] }, "execution_count": 156, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s4n.factor()" ] }, { "cell_type": "code", "execution_count": 157, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\pi {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} {y_H}^{2} + b^{2} {y_H}^{2} + 2 \\, a^{2} m - 2 \\, b^{2} m - a^{2} + b^{2}\\right)}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\pi {\\left(a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} {y_H}^{2} + b^{2} {y_H}^{2} + 2 \\, a^{2} m - 2 \\, b^{2} m - a^{2} + b^{2}\\right)}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}}$$" ], "text/plain": [ "-1/4*pi*(a^2*pf^2 - b^2*qf^2 - a^2*yH^2 + b^2*yH^2 + 2*a^2*m - 2*b^2*m - a^2 + b^2)/((2*yH^2 + 1)*sqrt(yH^2 + 1))" ] }, "execution_count": 157, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = s4n.factor()/s2d\n", "s" ] }, { "cell_type": "code", "execution_count": 158, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\pi {\\left(a^{2} {y_H}^{4} - b^{2} {y_H}^{4} + a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} + b^{2}\\right)}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\pi {\\left(a^{2} {y_H}^{4} - b^{2} {y_H}^{4} + a^{2} {\\mathfrak{p}}^{2} - b^{2} {\\mathfrak{q}}^{2} - a^{2} + b^{2}\\right)}}{4 \\, {\\left(2 \\, {y_H}^{2} + 1\\right)} \\sqrt{{y_H}^{2} + 1}}$$" ], "text/plain": [ "-1/4*pi*(a^2*yH^4 - b^2*yH^4 + a^2*pf^2 - b^2*qf^2 - a^2 + b^2)/((2*yH^2 + 1)*sqrt(yH^2 + 1))" ] }, "execution_count": 158, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s.subs({m: m_yH}).factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The denominator clearly agrees with Eq. (4.33)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Conclusion:** we have full agreement with Eq. (4.33)." ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.3", "language": "sagemath", "metadata": { "cocalc": { "description": "Open-source mathematical software system", "priority": 1, "url": "https://www.sagemath.org/" } }, "name": "sage-9.3", "resource_dir": "/ext/jupyter/kernels/sage-9.3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.2" } }, "nbformat": 4, "nbformat_minor": 4 }