# 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? Jun 5, 2015 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, 2015 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. Jun 7, 2015 at 16:07

Every cotangent bundle is a symplectic manifold and hence a Poisson manifold. Symplectic manifolds encode unconstrained mechanics. A symplectic manifold has only a single symplectic leaf, namely itself.

More general Poisson manifolds (often obtained by a process called symplectic reduction) encode already constrained mechanics: Each Casimir becomes a constraint when fixed at a constant value, which produces a Poisson homomorphism to the constrained Poisson manifold. The symplectic leaves of a Poisson manifold are the submanifolds obtained in this way by fixing the values of a complete collection of Casimirs, i.e., generating the family of all Casimirs. (Symplectic manifolds have no Casimirs apart from constants, which explains why they are their own symplectic leaf.)

To impose constraints not given by Casimirs (e.g., when imposing constraints on a symplectic space) one must first restrict the original Poisson algebra to the centralizer of the constraint (the set of functions whose Poisson bracket with the constraint set vanishes); in this centralizer (which may in certain cases be interpreted as the algebra of gauge invariant functions), the original constraints are Casimirs, and the above applies.

A good book about Poisson manifolds and their use in physics is the book ''Mechanics and symmetry" by Marsden and Ratiu.

• Thank you so much for this answer, a last point would be how does the Dirac bracket interplay with this structure? Is the Poisson bracket for the leaves and the Dirac bracket on the Poisson manifold? Thank you again!
– user58536
Jun 4, 2015 at 16:28
• The suggested book is exactly what I was looking for!
– user58536
Jun 4, 2015 at 17:17
• @JanetthePhysicist: The relation to the Dirac bracket is complicated - it amounts to giving a more explicit description of the centralizer. Jun 5, 2015 at 9:39