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I have a simple question concerning how to make a linear change of variables without destroying the symplectic structure of the Hamiltonian?

For example suppose I have a Hamiltonian in action-angle variables given by

\begin{eqnarray} \dot{\theta_1} &=& r_1 +r_2 \\ \dot{\theta_2} &=& r_1 \\ \dot{r_1} &=& \theta_2 \\ \dot{r_2} &=& \theta_1 \\ \end{eqnarray}

Then I want to define a new variable say $\rho= r_1 +r_2$. What will be the associated angle? I am pretty sure I cannot just pick $\theta_1$ or $\theta_2$

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this is not my specialty, but there is such a thing as symplectic diffeomorphisms (or change of variables) : en.wikipedia.org/wiki/Symplectomorphism –  Glougloubarbaki Jun 21 '12 at 14:27
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2 Answers 2

up vote 1 down vote accepted

With new action variables $R_1 = r_1 + r_2$ and $R_2 = r_2$, you can take the new angles as $\phi_1 = \theta_1$ and $\phi_2 = \theta_2 - \theta_1$. In this way you preserve the canonical one-form, $$R_1 \,d\phi_1 + R_2 \, d\phi_2 = (r_1 + r_2) \, d\theta_1 + r_2 \, (d\theta_2 - d\theta_1) = r_1 \, d\theta_1 + r_2 \, d\theta_2,$$ and therefore also the symplectic form. To learn more, read about canonical transformations and their generating functions.

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Any admissible hamiltonian change of coordinates must preserve the Poisson brackets

http://en.wikipedia.org/wiki/Poisson_bracket

On web the page of Giovanni Gallavotti contains a lot of materials (also free books) on the subject. At the moment it is not accessible.

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That wasn't my question. Suppose I have a transform for my action variables can I find a corresponding transformation for my angle variables such that I preserve the canonical form? –  Novo Jun 21 '12 at 17:01
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