Path: blob/main/17Manifold_One_Parameter.ipynb
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17. One-parameter groups of transformations
This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).
Let be a smooth manifold. A one-parameter group of transformations , on is a smooth map such that
ParseError: KaTeX parse error: Undefined control sequence: \label at position 31: …} ϕ(x, 0) = x, \̲l̲a̲b̲e̲l̲{}\tag{17.1} \e…ParseError: KaTeX parse error: Undefined control sequence: \label at position 86: …M, \ t, s ∈ R. \̲l̲a̲b̲e̲l̲{}\tag{17.2} \e…If we put
then , for is a smooth map and according to (17.2)
i.e.,
ParseError: KaTeX parse error: Undefined control sequence: \label at position 51: …t = ϕ_t ◦ ϕ_s, \̲l̲a̲b̲e̲l̲{}\tag{17.3} \e…since .
The map is the identity on since from (17.1) it follows for all . From (17.3) it follows , which means that each map has an inverse, , which is also smooth. Therefore, each is a diffeomorphism of onto itself. Thus, the set of transformations is an Abelian group of diffeomorphisms of onto , and the map is a homomorphism from the additive group of the real numbers into the group of diffeomorphisms of .
Example 17.1
Let us check that the formula
defines a one-parameter group of transformations i.e., the conditions (17.1), (17.2) are satisfied.
First let us check (17.1), i.e. that :
Now check that (17.2) holds for the first component of :
and for the second component of :
We have checked the condition (17.2).
Infinitesimal generator of
Each one-parameter group of transformations on defines a family of curves in . The map given by is a smooth curve in M for each . Since , the tangent vector to the curve at (defined in (8.9)) belongs to . The infinitesimal generator of is the vector field such that
The infinitesimal generator of is a vector field tangent to the curves generated by the one-parameter group of transformations.
Example 17.2
Let us compute the infinitesimal generator of the one-parameter group from the previous example in Cartesian coordinates of .
Thus the infinitesimal generator is
Example 17.3
Plot the infinitesimal generator from the previous example.
Example 17.4
For from Example 17.1, plot the curves for some selected points , where .
Example 17.5
Check that the formulas
define a one-parameter group of transformations.
Example 17.6
Compute components of the infinitesimal generator of from the previous example. Plot .
Thus
Example 17.7
For from Example 17.5 plot the curves for some selected points , where .
Example 17.8
Consider defined by
for fixed . Show that defines a one-parametric group of transformations.
Example 17.9
Compute components of the infinitesimal generator of from the previous example for .
Thus
Example 17.10
Plot the vector field from the previous example.
Example 17.11
For from Example 17.8 plot the curves for some selected points , where .
Example 17.12
Consider defined by
for fixed . Show that defines a one-parameter group of transformations.
Example 17.13
Compute components of the infinitesimal generator of from the previous example for .
Thus
Example 17.14
Plot the vector field from the previous example.
Example 17.15
For from Example 17.12 plot the curves for some selected points , where .
What's next?
Take a look at the notebook Integral curves.