Examen_2020_2_02_enunciado.ipynb

Kernel: Python 3

I) The period of a pendulum of length l oscillating at a large angle $\alpha$ is given by $$T=T_{0} \frac{\sqrt{2}}{\pi} \int_{0}^{\alpha} \frac{d \theta}{(\cos \theta-\cos \alpha)^{\frac{1}{2}}}$$ where $$T_{0}=2 \pi \sqrt{\frac{l}{g}}$$ is the period of the same pendulum at small amplitudes. Any numerical evaluation of the integral as is would fail (explain why). If we change the variable by writing: $$\sin \frac{\theta}{2}=\sin \frac{\alpha}{2} \sin \phi$$ we can get: $$T=\frac{2 T_{0}}{\pi} \int_{0}^{\frac{\pi}{2}} \frac{d \phi}{\left(1-\sin ^{2} \frac{\alpha}{2} \sin ^{2} \phi\right)^{\frac{1}{2}}}$$ which is a well-behaved integral. Write a program to use the above integral to calculate the ratio $T/T_0$ for integral amplitudes $0^\circ \le\alpha \le 90^\circ$. Output these values as a pandas dataframe showing the amplitude in degrees and radians as well as $T/T_0$, and make a plot with the two columns. Explain the result when $\alpha = 0.$

II)

Casas-Ibarra parameterization

Consider a $n\times n$ symmetric matrix $A$. We can assumme without lost of generality that this can be generated from a matrix $Y$ such that $$A=Y^{\operatorname{T}}Y$$ Theorem 1 gurantees that exists an ortogonal matrix $U$ such that $$U^{\operatorname{T}} A U=U^{\operatorname{T}} Y^{\operatorname{T}}Y U=D_\lambda$$ where $$D_{\lambda}=A_{\text{diag}}=\operatorname{diag}\left(\lambda_1,\lambda_2,\ldots,\lambda_n\right)$$ where $\lambda_i$ are the eigenvalues of $A$. Therefore $$\begin{align} Y^{\operatorname{T}}Y =&U D_\lambda U^{\operatorname{T}}\\ =&U D_{\sqrt{\lambda}} D_{\sqrt{\lambda}} U^{\operatorname{T}}\\ \end{align}$$ where $$D_{\sqrt{\lambda}}=\operatorname{diag}\left(\sqrt{\lambda_1},\sqrt{\lambda_2},\ldots \sqrt{\lambda_n}\right)$$ Therefore, exists an ortogonal arbitrary matrix $R$, such that $$Y^{\operatorname{T}}Y =U D_{\sqrt{\lambda}}R^{\operatorname{T}}R D_{\sqrt{\lambda}} U^{\operatorname{T}}\\$$

In this way, the matrix $Y$ can be parameterized in terms of $R$ as $$Y=R D_{\sqrt{\lambda}} U^{\operatorname{T}}$$

1. By using the previous equations, build a matrix $Y$ $2\times 2$ with the following conditions

• $R$ is an orthogonal matrix with a mixing angle as a random number between $(0,2\pi)$. Use your identification number as the seed of the random number generator.

• The eigenvalues are $\lambda_1=2$ and $\lambda_2=4$.

• $U$ is a diagonalization matrix with mixing angle $\pi/4$

2. Build the matrix $A$ and check that has the proper eigenvalues and eigenvectors