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 where {fi, fj} = 0.
(In other words, they are pairwise in involution.)
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 As a direct application we have the following.
there is a symplectic chart such that one of its coordinates is H.
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