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Lie-Poisson and Euler-Poincaré Reduction

  • Jerrold E. Marsden
  • Tudor S. Ratiu
Chapter
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Part of the Texts in Applied Mathematics book series (TAM, volume 17)

Abstract

Besides the Poisson structure on a symplectic manifold, the Lie-Poisson bracket on g*, the dual of a Lie algebra, is perhaps the most fundamental example of a Poisson structure. We shall obtain it in the following manner. Given two smooth functions F, HF(g*), we extend them to functions F L , H L (respectively, F R , H R ) on all T*G by left (respectively, right) translations. The bracket {F L , H L } (respectively, {F R , H R }) is taken in the canonical symplectic structure Ω on T*G. The result is then restricted to g* regarded as the cotangent space at the identity; this defines {F, H}. We shall prove that one gets the Lie-Poisson bracket this way. This process is called Lie-Poisson reduction. In §14.6 we show that the symplectic leaves of this bracket are the coadjoint orbits in g*.

Keywords

Variational Principle Poisson Bracket Integral Curve Poisson Structure Left Translation 
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Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Jerrold E. Marsden
    • 1
  • Tudor S. Ratiu
    • 2
  1. 1.California Institute of TechnologyControl and Dynamical Systems, 107-81PasadenaUSA
  2. 2.Département de mathématiquesEcole polytechnique fédérale de LausanneLausanneSwitzerland

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