Poisson Manifolds

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


The dual g* of a Lie algebra g carries a Poisson bracket given by
$$ \left\{ {F,G} \right\}\left( \mu \right) = \left\langle {\mu \left[ {\frac{{\delta F}}{{\delta \mu ,}}\frac{{\delta G}}{{\delta \mu }}} \right]} \right\rangle $$
for μ∈ g*, a formula found by Lie, [1890, Section 75]. As we saw in the Introduction, this Lie-Poisson bracket description of many physical systems. This bracket is not the bracket associated with any symplectic structure on g*, but is an example of the more general concept of a Poisson manifold. On the other hand, we do want to understand how this bracket is associated with a symplectic structure on coadjoint orbits and with the canonical symplectic structure on T* G.These facts are developed in Chapters 13 and 14. Chapter 15 shows how this works in detail for the rigid body.


Poisson Bracket Symplectic Form Symplectic Manifold Poisson Structure Coadjoint Orbit 
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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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