Lie-Poisson and Euler-Poincaré Reduction

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


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*.


Variational Principle Poisson Bracket Integral Curve Poisson Structure Left Translation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations