Mathematical Notes

, Volume 78, Issue 3–4, pp 297–303 | Cite as

On the Linearization of Hamiltonian Systems on Poisson Manifolds

  • Yu. M. Vorob'ev


The linearization of a Hamiltonian system on a Poisson manifold at a given (singular) symplectic leaf gives a dynamical system on the normal bundle of the leaf, which is called the first variation system. We show that the first variation system admits a compatible Hamiltonian structure if there exists a transversal to the leaf which is invariant with respect to the flow of the original system. In the case where the transverse Lie algebra of the symplectic leaf is semisimple, this condition is also necessary.

Key words

Hamiltonian system Poisson bracket linearization Poisson coupling normal bundle first variation system Hamiltonian vector field 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Yu. M. Vorob'ev
    • 1
    • 2
  1. 1.University of SonoraMexico
  2. 2.Moscow Institute of Electronics and MathematicsMoscowRussia

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