Flat Bi-Hamiltonian Structures and Invariant Densities

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A bi-Hamiltonian structure is a pair of Poisson structures P, Q which are compatible, meaning that any linear combination αP+ βQ is again a Poisson structure. A bi-Hamiltonian structure (P, Q) is called flat if P and Q can be simultaneously brought to a constant form in a neighborhood of a generic point. We prove that a generic bi-Hamiltonian structure (P, Q) on an odd-dimensional manifold is flat if and only if there exists a local density which is preserved by all vector fields Hamiltonian with respect to P, as well as by all vector fields Hamiltonian with respect to Q.

Original languageEnglish (US)
Pages (from-to)1415-1427
Number of pages13
JournalLetters in Mathematical Physics
Issue number10
StatePublished - Oct 1 2016


  • Casimir functions
  • bi-Hamiltonian structures
  • invariant densities

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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