Carathéodory–Jacobi–Lie theorem

The Carathéodory–Jacobi–Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.

Statement
Let M be a 2n-dimensional symplectic manifold with symplectic form &omega;. For p ∈ M and r &le; n, let f1, f2, ..., fr be smooth functions defined on an open neighborhood V of p whose differentials are linearly independent at each point, or equivalently


 * $$df_1(p) \wedge \ldots \wedge df_r(p) \neq 0,$$

where {fi, fj} = 0. (In other words, they are pairwise in involution.) Here {–,–} is the Poisson bracket. Then there are functions fr+1, ..., fn, g1, g2, ..., gn defined on an open neighborhood U ⊂ V of p such that (fi, gi) is a symplectic chart of M, i.e., &omega; is expressed on U as


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

Applications
As a direct application we have the following. Given a Hamiltonian system as $$(M,\omega,H)$$ where M is a symplectic manifold with symplectic form $$\omega$$ and H is the Hamiltonian function, around every point where $$dH \neq 0$$ there is a symplectic chart such that one of its coordinates is H.