I'm wondering when the phase space of a Hamiltonian system looses its symplectic structure. I think it happens when the Hamiltonian $H$ depends on a set of other variables $S_1,...,S_k$ as well as on the set of canonical variables $q_i, p_i$.
So, besides having $$\begin{gather} \dot{q}_i = \bigl\{ q_i, H \bigr\} = \frac{\partial H}{\partial p_i}\\ \dot{p}_i= \bigl\{ p_i, H \bigr\} = - \frac{\partial H}{\partial q_i} \end{gather}$$ we have also some $\dot{S}_i = \bigl\{ S_i, H \bigr\}$, which is not a Hamilton equation.
Is it right? Could anyone make this clearer?


1 Answer 1

  1. Well, more generally, a Hamiltonian system $\dot{z}^I=\{z^I,H\}$ with a Hamiltonian function $H:M\times \mathbb{R}\to \mathbb{R}$ is defined on a (not necessarily invertible) Poisson manifold rather than a symplectic manifold.

  2. A (not necessarily invertible) Poisson manifold might not have local canonical/Darboux coordinates, cf. OP's example.

  3. An important example of a non-invertible Poisson bracket is the Dirac bracket for constrained systems.


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