Poisson Manifolds

Abstract

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.