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, ..., f_r 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 {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

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

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.

References

• Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.