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.