Английская Википедия:Carathéodory–Jacobi–Lie theorem
Шаблон:Short description 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
- <math>df_1(p) \wedge \ldots \wedge df_r(p) \neq 0,</math>
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
- <math>\omega = \sum_{i=1}^n df_i \wedge dg_i.</math>
Applications
As a direct application we have the following. Given a Hamiltonian system as <math>(M,\omega,H)</math> where M is a symplectic manifold with symplectic form <math>\omega</math> and H is the Hamiltonian function, around every point where <math>dH \neq 0</math> there is a symplectic chart such that one of its coordinates is H.
References
Шаблон:Differential-geometry-stub
