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Darboux's lemma once more


Author: Hans Samelson
Journal: Proc. Amer. Math. Soc. 123 (1995), 1253-1255
MSC: Primary 58A10; Secondary 53C15
DOI: https://doi.org/10.1090/S0002-9939-1995-1246536-3
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.


References [Enhancements On Off] (What's this?)

  • [1] C. Carathéodory, Variartionsrechnung und partielle Differentialgleichungen erster Ordnung, Teubner, Leipzig, 1935, p. 124; English transl., Calculus of variations and partial differential equations of first order, Holden-Day, San Francisco, 1965, p. 125.
  • [2] J. Moser, On the volume elements of a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. MR 0182927 (32:409)
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  • [4] A. Weinstein, Symplectic structures on Banach manifolds, Bull. Amer. Math. Soc. 75 (1969), 1040-1041. MR 0245052 (39:6364)
  • [5] E. T. Whittaker, A treatise on the analytical dynamics of particles and rigid bodies, Dover, New York, 1944, p. 317. MR 0010813 (6:74d)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1246536-3
Keywords: Canonical coordinates, Hamiltonian transformation theory
Article copyright: © Copyright 1995 American Mathematical Society

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