# Using canonical transform to show area preserving

• bagram
In summary, the conversation discusses how to prove that a given Poincare map is area preserving for all values of r. The suggested method involves looking at the Jacobian of the transformation and checking if it equals 1. While the presence of I_n makes the map nonlinear, it is still possible to use this method to check for area preservation.

## 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?

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

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

Yes. It's a little tedious.