Using canonical transform to show area preserving

[bagram]

Homework Statement

Given a certain poincare map, show that it is area preserving for all values of r

Homework Equations

$$\binom{x_{n+1}}{y_{n+1}}=\begin{pmatrix}e^{r} & 0 \\ 0 & e^{-r} \end{pmatrix}\begin{pmatrix}cos(\phi+I_{n}) & -sin(\phi+I_{n}) \\ sin(\phi+I_{n}) & cos(\phi+I_{n}) \end{pmatrix}\begin{pmatrix}x_{n}\\ y_{n} \end{pmatrix}$$

where
$$I_{n}=x_{n}^2+y_{n}^2$$

The Attempt at a Solution

I tried to prove that the determinant $$\binom{x_{n+1}}{y_{n+1}}=\begin{pmatrix}e^{r} & 0 \\ 0 & e^{-r} \end{pmatrix}\begin{pmatrix}cos(\phi+I_{n}) & -sin(\phi+I_{n}) \\ sin(\phi+I_{n}) & cos(\phi+I_{n}) \end{pmatrix}$$ is equal to 1, I believe that is wrong since the I in the matrix is X^2+Y^2, so I was wondering should i use the Poisson bracket method?

 

[TSny]

The presence of $I_n$ makes the map nonlinear, so you can't just look at the determinant of the map to see if it is area preserving.

Think of the map as being in the form $$x_{n+1} = f(x_n,y_n)$$ $$y_{n+1} = g(x_n,y_n)$$
Then consider the Jacobian of this transformation.
See for example http://www.math.harvard.edu/archive/118r_spring_05/handouts/henon.pdf

 

[bagram]

Ahhh okay, so then i just do the partial of each entry in the matrix to check if the det=1?

 

[TSny]

Yes. It's a little tedious.

 

[paranormal]

Yes, using the Poisson bracket method would be a good approach to show that the given poincare map is area preserving. The Poisson bracket is a mathematical tool used to study the dynamics of systems and is particularly useful in Hamiltonian mechanics, which deals with systems that conserve energy. In this case, the poincare map can be seen as a Hamiltonian system, where the phase space coordinates x and y are analogous to position and momentum, and the parameter r is analogous to energy.

To show that the poincare map is area preserving, you can use the fact that the Poisson bracket of two functions is equal to the negative of their respective partial derivatives with respect to the phase space variables. In this case, the functions are the coordinates x and y, and the Poisson bracket is given by:

{f,g} = \frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}

If you compute the Poisson bracket of x and y, you will find that it is equal to 1, which means that the area of the phase space is preserved under the poincare map. This is because the Poisson bracket of two functions is a measure of how much one function changes when the other is varied, and in this case, the coordinates x and y do not change under the poincare map, so the Poisson bracket is equal to 1. This proves that the poincare map is area preserving for all values of r.

In summary, using the Poisson bracket method, we can show that the poincare map is area preserving by computing the Poisson bracket of the phase space coordinates x and y and showing that it is equal to 1. This is a more rigorous and general approach compared to trying to show that the determinant of the transformation matrix is equal to 1, which only holds for a specific value of r.