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The Hamiltonian structure associated to evolution-type Lagrangians

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Abstract

It is demonstrated that, for a certain class of Lagrangians, which includes those for the Korteweg-de Vries (KdV) hierarchy, the Hamiltonian structure provided by the Hamilton-Cartan formalism is precisely the one discovered by Gardner for the KdV equation. A simple geometric relation between the Cartan 2-forms for this class of Lagrangians and the Cartan 1-forms for the associated stationary problems is given. This relation provides a new proof of the theorem of Bogoyavlenski-Novikov and Gel'fand-Dikii on the integrability of the stationary Korteweg-de Vries equations.

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References

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Authors and Affiliations

  1. Institute for Advanced Study, Princeton, NJ, U.S.A.

    W. F. Shadwick

  2. Applied Mathematics Department, University of Waterloo, Waterloo, Ontario, Canada

    W. F. Shadwick

Authors
  1. W. F. Shadwick
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Research supported by the Natural Sciences and Engineering Research Council of Canada.

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Shadwick, W.F. The Hamiltonian structure associated to evolution-type Lagrangians. Lett Math Phys 6, 271–276 (1982). https://doi.org/10.1007/BF00400321

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  • DOI: https://doi.org/10.1007/BF00400321

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