Journal of Dynamics and Differential Equations

, Volume 6, Issue 1, pp 53–69

Completely integrable bi-Hamiltonian systems

  • Rui L. Fernandes

DOI: 10.1007/BF02219188

Cite this article as:
Fernandes, R.L. J Dyn Diff Equat (1994) 6: 53. doi:10.1007/BF02219188


We study the geometry of completely integrable bi-Hamiltonian systems and, in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that under some natural hypothesis, such a structure exists in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation, relative to the affine structure determined by the action variables. This generalizes a result of Brouzet for dimension four.

Key words

Bi-Hamiltonian system completely integrable system 

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Rui L. Fernandes
    • 1
    • 2
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolis
  2. 2.Departamento de MatemáticaInstituto Superior TécnicoLisbonPortugal

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