For classical systems we can define a configuration manifold, whose cotangent bundle is a momentum phase space equipped with a closed, non-degenerate 2-form. Upon the commutative algebra of smooth functions on the cotangent bundle we define a bilinear bracket that endows the set with a Lie algebra structure. We say that a Poisson manifold is exactly this description, but is regularly foliated with symplectic leaves, (symplectic sub manifolds). I am interested in constrained dynamics.
I am struggling to understand how I should think of these leaves,
- are they each the cotangent bundles to configuration manifolds, equipped with coordinates that describe time slices? Or is the Poisson manifold itself the cotangent bundle to the configuration manifold?
In Arnold's text on classical mechanics he identifies the symplectic manifolds of integrable systems with the torus. Should these be viewed as the submanifolds or the Poisson manifold? Unification of this question my first would help me very much!