Darboux’s lemma once more

Abstract:

Darboux’s lemma states that a closed nondegenerate two-form $\Omega$, defined on an open set in ${\mathbb {R}^{2n}}$ (or in a 2n-dimensional manifold), can locally be given the form $\sum {d{q_i} \wedge d{p_i}}$, in suitable coordinates, traditionally denoted by ${q_1},{q_2}, \ldots ,{q_n},{p_{1,}}{p_2}, \ldots ,{p_n}$. There is an elegant proof by J. Moser and A. Weinstein. The author has presented a proof that was extracted from Carathéodory’s book on Calculus of Variations. Carathéodory works with a (local) "integral" of $\Omega$, that is, with a one-form $\alpha$ satisfying $d\alpha = \Omega$. It turns out that the proof becomes much more transparent if one works with $\Omega$ itself.