Why is every Hamiltonian system locally integrable?

It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the $2n$-dimensional symplectic manifold on which the Hamiltonian vector field is defined, it is possible to find $n$ integrals of motion in involution.

In general these do not globally extend to give the compact Lagrangian fibration/foliation appearing in the Arnold-Liouville theorem, but here I want to focus on this purely local situation.

What is the simplest rigorous argument you can give me of this fact?

• Darboux's theorem? – Fan Zheng Jan 21 '17 at 0:19
• I fail to see how the Hamiltonian vector field and its symmetries take part in Darboux theorem. What I am looking for is the result that, locally, I can find a Lagrangian foliation tangent to the vector field. Why does this come trivially from the normal form of the symplectic structure? – issoroloap Jan 21 '17 at 7:36
• @issoroloap I have a feeling that Fan Zheng may have a point. If you look at the proof of Darboux's theorem in Arnold's book on mechanics, the symplectically "flat" coordinates there seem to have been constructed exactly by finding the local Hamitonians in involution that generate the local tangent Lagrangian submanifold. – Futurologist Feb 13 '17 at 5:47

For non-singular Hamiltonian systems, you can see it as a consequence of a slight generalization of the Darboux Theorem which is known as the Carathéodory--Jacobi--Lie Theorem (see, e.g. Libermann, Marle, Symplectic Geometry and Analytical Mechanics).

Carathéodory--Jacobi--Lie Theorem Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold, with associated Poisson bracket $\{-,-\}$. Assume to have, for some $0\leq k\leq n$, smooth functions $f_1,\ldots,f_k$, defined on a neighborhood $U$ of a point $x$ in $M$, which are independent and Poisson-commuting, i.e.

$$df_1\wedge\ldots\wedge df_k\neq 0,$$

$$\{f_i,f_j\}=0,\text{ for all }0\leq i<j\leq k.$$

Then there exist smooth functions $f_{k+1},\ldots,f_n,g_1,\ldots,g_n$ defined an open neighborhood $V$ of $x$ in $U$ such that

$$\omega=\sum_{i=1}^n df_i\wedge dg_i.$$

I hope it helps.

• yes, thank you! I had JUST found this theorem myself, precisely in the reference you give. I agree that it is a slight generalization of Darboux Theorem, when you look at the proof. However I am glad I actually learned it. Thanks again. – issoroloap Jan 21 '17 at 8:53
• This only proves the result away from critical points of the Hamiltonian. – Ben McKay Jan 21 '17 at 9:26
• @Ben McKay according with your comment I have now edited the answer to make explicit the underlying assumption of non-singularity on the Hamiltonian system – agtortorella Jan 21 '17 at 11:58
• thanks @Ben, yes you are right of course, I even edited the question. – issoroloap Jan 21 '17 at 13:03