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.
Key wordsHamiltonian system Poisson bracket linearization Poisson coupling normal bundle first variation system Hamiltonian vector field
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- 1.J. Marsden, T. Ratiu, and G. Rangel, “Symplectic connections and the linearization of hamiltonian systems,” Proc. Roy. Soc. Edinburgh. Sect. A, 117 (1991), 329–380.Google Scholar
- 2.J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, New York, 1994.Google Scholar
- 3.M. V. Karasev and Yu. M. Vorobjev, “Adapted connections, Hamiltonian dynamics, geometric phases, and quantization over isotropic submanifolds,” Amer. Math. Soc. Transl., 187 (1998), no. 2, 203–326.Google Scholar
- 4.R. Flores Espinoza and Yu. M. Vorobiev, “Hamiltonian formalism for fiberwise linear systems,” Bol. Soc. Mat. Mexicana, 6 (2000), no. 3, 213–234.Google Scholar
- 5.Yu. Vorobiev, “Hamiltonian equations of the first variation equations,” Mathematics, 191 (2000), no. 4, 447–502.Google Scholar
- 6.Yu. Vorobjev, “Coupling tensors and Poisson geometry near a single symplectic leaf,” in: Lie Algebroids, vol. 54, Banach Center Publ., Warszawa, 2001, pp. 249–274.Google Scholar
- 7.Yu. M. Vorob'ev, “On linearized Poisson structures,” Mat. Zametki [Math. Notes], 70 (2001), no. 4, 486–493.Google Scholar
- 8.A. Weinstein, “The local structure of Poisson manifolds,” J. Diff. Geom., 18 (1983), 523–557.Google Scholar
- 9.A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Math. Lecture Notes, Amer. Math. Soc., Providence, RI, 1999.Google Scholar
- 10.M. V. Karasev and Yu. M. Vorobjev, “Deformations and cohomology of Poisson manifolds,” in: Lecture Notes in Math., vol. 1453, Springer-Verlag, Berlin, 1990, pp. 271–289.Google Scholar