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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Darboux's lemma once more

Author: Hans Samelson
Journal: Proc. Amer. Math. Soc. 123 (1995), 1253-1255
MSC: Primary 58A10; Secondary 53C15
MathSciNet review: 1246536
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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.

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Keywords: Canonical coordinates, Hamiltonian transformation theory
Article copyright: © Copyright 1995 American Mathematical Society