A free pivoted rigid body motion is described by the Euler’s equations. This can be put in Hamiltonian form in an odd-dimensional Poisson manifold using a Lie-Poisson bracket, but it also admits a standard symplectic description. Are there examples of physical system which do not admit a symplectic description?
Dear Florin, Any physical systems could be associated with a configuration space and its cotangent bundle where the symplectic structures (and phase space) live, Hence we can recover a symplectic manifold (and forms) to any manifold and consequently any physical problems, However the reverse may not be correct, i.e. the symplectic manifolds which are not a cotangent bundle M* of some M , e.g. symplectic manifold without boundaries.
Thank you Manouchehr, but this is not answering my question. Poisson manifolds can arise out of symplectic ones by symplectic reduction. But how about the reverse?
Dear Florin, My post was aimed at your last question " Are there examples of physical system which do not admit a symplectic description?"
There do exist physical systems, that don't have a Hamiltonian description. Examples are physical systems that are not invariant under time translations-there isn't any Hamiltonian. Typical examples are systems where dissipative effects compete with forcing terms. Now one can ask, of course, whether it is possible to ``close'' the system, i.e. to provide a model of the additional degrees of freedom that would be necessary to lead to a Hamiltonian description of the full system, while preserving the properties of the subsystem one started with. I don't think there's a general procedure for this, but it might be useful to ask about universality classes.
Dear Florin, I am not familiar with physical description of symplectic and Poisson manifolds. Although any Poisson manifold has a symplectic foliation, there are Poisson manifolds that are not symplectic. The most typical example is the linear Poisson structure on the dual of a Lie algebra, the Poisson tensor is linear, thus identically null n zero so not invertible!
For an odd dimensional manifold $X $ which is the total space of a principal bundle over a Poisson manifold, on can construct a Lie bracket structure on the space of smooth functions on $X$, but I am not sure that this bracket satisfies the Leibniz rule.
Please see the remark in the last part of the following MO post
https://mathoverflow.net/questions/284192/symplectic-principal-bundles
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