# What exactly is the relationship between the symplectic 2-form and the frequency of leaves of integrable systems in classical mechanics?

In classical mechanics we equip a differential manifold with a closed symplectic 2-form $\omega$. The symplectic leaves of integrable systems also have a unique frequency, in literature denoted $\omega$. In both cases we talk of "degeneracies" in frequencies and the 2-form.

My question is simple yet one that I am reluctant to dismiss.

1. Are these two expressions of the same object?

My intuition says absolutely not, no. At no point in literature have I seen their equivalence however they appear to dictate similar properties and are both important to presymplectic manifolds and discussions of stability of manifolds in time. Mathematically they seem a world apart, but I just wanted some re-assurance I suppose!

In Darboux charts we can write the 2-form as $\omega =dq^i\wedge dp_i$, and the frequency as $\dot \theta =\partial H/\partial I=\omega$, where $\theta$ and $I$ are action-angle variables for a given torus.

1. If they are not the same object, can their properties be linked?

For example if we had a degeneracy then the 2-form would have zero-valued eigenvalues, the frequency would have a linear dependance between each angle coordinate (small denominators). In addition we can label a torus with both its frequency and its 2-form.

If anyone had expertise on this I would be grateful for a little reassurance!

EDIT

A 2-form $\omega$ on a differentiable manifold $M$ is presymplectic if it closed but not necessarily non-degenerate, the pair $(M,\omega)$ make a presymplectic manifold. The characteristic foliation of a presymplectic manifold is closely tied with the integrability through Frobenius theorem. And we always talk about integrability when discussing the frequencies we talk of above.

As I say mathematically I have seen nothing that links them, nor have I seen any discussion of them being linked in any way, and I agree how mathematically can a 2-form be anything but a 2-form.

• But the close ties with integrability and foliations as well as degeneracies is disconcerting for me to outright dismiss there being something.

• In addition we label a foliation with a frequency and we define a 2-form on each foliation, as well as repeated use of the same symbol (trivial I know).

• If one is a 2-form and the other is not, how can they be the same object? – MBN Jun 17 '15 at 12:46
• Comments to the question (v5): Which literature discusses [any of above matters] in the context of presymplectic manifolds? – Qmechanic Jun 17 '15 at 13:56
• To me it doesn't seem that these can be linked... Let's say I write $L = \omega_{ab}X^a \wedge X^b - H(X^a)$ for phase space variables $X^a$ ($a=1,\cdots,2N$). I can always put $\omega$ into Darboux form at least locally, so $L = p_i \wedge q^i - H(p_i,q^i)$ where $i=1,\cdots,N$. But the procedure that puts $\omega$ into Darboux form is independent of the form of the Hamiltonian. Said differently, once I'm in Darboux form, I can pick any Hamiltonian I want, and different Hamiltonians will lead to different action angles / frequencies. Does that make sense, or am I missing something basic? – Andrew Jun 22 '15 at 20:12