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 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
where = 0. 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 is a symplectic chart of M, i.e., ω is expressed on U asApplications
As a direct application we have the following. Given a Hamiltonian system as where M is a symplectic manifold with symplectic form and H is the Hamiltonian function, around every point where there is a symplectic chart such that one of its coordinates is H.
