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MathNeophyte
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- 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 x0
$$x(\tau) = \Phi(x_{0},\tau)$$ 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 x0
$$x(\tau) = \Phi(x_{0},\tau)$$ 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?