Carathéodory–Jacobi–Lie theorem

Statement

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

where {f_i, f_j} = 0. (In other words they are pairwise in involution.) Here {–,–} is the Poisson bracket. Then there are functions f_r+1, ..., f_n, g₁, g₂, ..., g_n defined on an open neighborhood U ⊂ V of p such that (f_i, g_i) is a symplectic chart of M, i.e., ω is expressed on U as

Applications

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