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.
- 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.
- 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!
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).