Poisson Manifolds

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

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.

Keywords

Poisson Bracket Symplectic Form Symplectic Manifold Poisson Structure Coadjoint Orbit
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.