The inspiration for symplectic geometry was from Hamiltonian mechanics. However, I am wondering how close the ties are between arbitrary symplectic manifolds and real physical systems.

In particular, it is my understanding that that the symplectic manifolds usually of interest to physicists are cotangent bundles of configuration spaces (but I am a mathematician so please correct me here if this is not correct). However, not every symplectic manifold is a cotangent bundle (for instance, the 2-sphere or the torus).

Question: For any given symplectic manifold $(M,\omega)$, is there a classical mechanical system which has $(M,\omega)$ as its phase space?

Particular examples of non cotangent bundle symplectic manifolds corresonding to real mechanical systems would also be useful.


2 Answers 2


There are lots of examples of singular phase spaces of mechanical systems when there is the possibility of locking, meaning that in certain regions of the configuration space, not all the degrees of freedom are available. These spaces are not even manifolds, let along cotangent bundles, although they are locally cotangent bundles (like all symplectic manifolds). Dynamics on them is pretty interesting. For instance you might ask if they are thermodynamically likely to lock or unlock.


$(M,\omega)$ is the phase space of a mechanical system if we can find a sufficiently regular scalar field $H$ on $M$. For if it exists, we can then define the associated Hamiltonian flow via

$$X_H := \iota_\omega\text dH$$

So the problem is now whether given the symplectic manifold $(M,\omega)$ we can always find at least one non-trivial differentiable scalar field defined on it to serve as a Hamiltonian.


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