# Symplectic leaves, tori and Poisson manifolds

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,

1. 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!

• Comment: This question (v3) seems too broad, cf. e.g. this recent meta post. OP asks about Nambu-Poisson (NP) structures and accepts an answer that doesn't seem to address NP structures. Perhaps remove the NP parts of the question, no? – Qmechanic Jun 5 '15 at 13:51
• @Qmechanic I agree that the answer doesn't address part of my question, but there doesn't seem to be much interest so I thought I would accept it since it helped me with a topic I am really struggling with. If another answer comes along I will offer another bounty of 50 points. Maybe I should have said that, but at the minute Im just so thankful to anyone who is willing to give up a little time that I felt it was worth it! – user58536 Jun 5 '15 at 15:26
• Qmechanic's suggestion to remove the nambu part is sensible - you can pose the removed part as a new question. It is never good to ask too much and too disparate things in one question. – Arnold Neumaier Jun 7 '15 at 16:07