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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 2''n''-dimensional symplectic manifold with symplectic form ω. For ''p'' ∈ ''M'' and ''r'' ≤ ''n'', let ''f''1, ''f''2, ..., ''f''r be smooth functions defined on an open neighborhood ''V'' of ''p'' whose differentials are linearly independent at each point, or equivalently
:
where = 0. (In other words they are pairwise in involution.) Here is the Poisson bracket. Then there are functions ''f''r+1, ..., ''f''n, ''g''1, ''g''2, ..., ''g''n 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
:
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