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I read a proof of Liouville’s theorem on integrable Hamiltonian systems. According to the theorem, an autonomous Hamiltonian system can be integrated in quadratures, given $n$ involutive first integrals $F_j(q_1\dots q_n,p_1\dots p_n)=a_j,\quad j=1\dots n$, if the Jacobian $\frac{D(F_1,\dots,F_n)}{D(p_1,\dots,p_n)}$ is non zero.

Since $\frac{D(F_1,\dots,F_n)}{D(p_1,\dots,p_n)}\not=0$, we can solve the system $F_j(q_1\dots q_n,p_1\dots p_n)=a_j,\quad j=1\dots n$ for $p$. That is $p_j=f_j(q_1,\dots,q_n,a_1,\dots,a_n)$. According to the proof $\sum_{j=1}^nf_jdq_j$ is an exact form and its potential function is Hamilton's characteristic function for the Hamiltonian system.

If we wanted to show existence of the solution, we may have referred to the general formula for existence of solutions to ordinary differential equations. It is understandable then why I perceive Liouville's theorem as a bit of a let down, for the proof is still not constructive. It only reduces the problem to integrating the above differential form.

What are the theorem's implications? How is it useful? Can we always find a potential function?

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Cross-posted from math.stackexchange.com/q/628522/11127 Related post by OP: physics.stackexchange.com/q/92485/2451 Related: physics.stackexchange.com/q/44576/2451 and links therein. –  Qmechanic Jan 9 at 21:03

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