Carathéodory–Jacobi–Lie theorem

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

Statement[edit]

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[edit]

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[edit]

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