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 $\theta_2$