On the linearization of Hamiltonian systems on poisson manifolds

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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.

Original languageEnglish
Pages (from-to)297-303
Number of pages7
JournalMathematical Notes
Issue number3-4
StatePublished - Sep 2005

Bibliographical note

Funding Information:
The author is very grateful to M. V. Karasev and R. Flores Espinoza for helpful discussions of various aspects of the present work. This research was supported in part by CONACYT under grant no. 35212-E.


  • First variation system
  • Hamiltonian system
  • Hamiltonian vector field
  • Linearization
  • Normal bundle
  • Poisson bracket
  • Poisson coupling


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