Communications in Mathematical Physics

, Volume 180, Issue 3, pp 529–586 | Cite as

Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures

  • Oleg I. Bogoyavlenskij


This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldMn. The central objects in this theory are supplementary invariant Poisson structuresPc which are incompatable with the original Poisson structureP1 for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP1 andP2 in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP1 nor with respect toP2. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.


Manifold Hamiltonian System Smooth Manifold Poisson Structure Celestial Mechanic 
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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Oleg I. Bogoyavlenskij
    • 1
  1. 1.Department of Mathematics and StatisticsQueen's UniversityKingstonCanada

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