Английская Википедия:Carathéodory–Jacobi–Lie theorem

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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, ..., f_r 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 {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

<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:Cite book
• Шаблон:Cite book

Шаблон:Differential-geometry-stub