- #1

- 1

- 1

- TL;DR Summary
- Trying to understand under what conditions the tautological/Liouville one form is integrable in the sense of having $n$ Poisson commuting integrals.

Apologies for potentially being imprecise and clunky, but I'm trying understand integrability of the following Hamiltonian

$$H(x,p)=\langle p,f(x) \rangle$$

on a 2n dimensional vector space

$$T^{\ast}\mathcal{M} =\mathbb{R}^{2n}.$$

Clearly this is just the 1-form $$\theta_{(x,p)} = \sum_{i}{p_i{}dq^{i}}.$$

I assume that anywhere along the flow of x given some x

$$x(\tau) = \Phi(x_{0},\tau)$$ the Hamiltonian is non-degenerate i.e. the Hessian of the Hamiltonian is full rank.

My questions are:

$$H(x,p)=\langle p,f(x) \rangle$$

on a 2n dimensional vector space

$$T^{\ast}\mathcal{M} =\mathbb{R}^{2n}.$$

Clearly this is just the 1-form $$\theta_{(x,p)} = \sum_{i}{p_i{}dq^{i}}.$$

I assume that anywhere along the flow of x given some x

_{0}$$x(\tau) = \Phi(x_{0},\tau)$$ the Hamiltonian is non-degenerate i.e. the Hessian of the Hamiltonian is full rank.

My questions are:

- Is H(x,p) Liouville integrable along the orbits of x given the above non-degeneracy assumption? I.e. Does this Hamiltonian have n, non-constant integrals that are commuting with respect to the cannonical Poisson bracket?
- How does this relate to the differential form version of the Frobenius theorem?