# Is there a physical interpretation of symplectic manifolds which are not cotangent bundles?

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.

• Subset/possible duplicate: physics.stackexchange.com/q/126676/50583 Jul 20, 2018 at 20:41
• Jul 20, 2018 at 20:49
• @ACuriousMind I did see that post; however, it is indeed only a subset of my more general question. In addition, the highest rated answer is rather trivial since it just reduces to the configuration space since the particle is always fixed. I was hoping for some more interesting examples. Several of the lower rated answers relate to quantum mechanics which is explicitly not what I was asking about (classical mechanical systems). But thank you for linking that post! Jul 20, 2018 at 21:03

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.