# Carathéodory–Jacobi–Lie theorem

The CarathéodoryJacobiLie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.

## Statement

Let M be a 2n-dimensional symplectic manifold with symplectic form ω. For p ∈ M and r ≤ 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., ω 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.