Poisson Manifolds

Abstract

In 1809 Poisson (see [144]) introduced a bracket on smooth functions, defined on R ²ⁿ, by the formula

$$\{ F,G\} : = \sum\limits_{i = 1}^n {\left( {\frac{{\partial F}}{{\partial {q_i}}}\frac{{\partial G}}{{\partial {p_i}}} - \frac{{\partial G}}{{\partial {q_i}}}\frac{{\partial F}}{{\partial {p_i}}}} \right)} .$$

(3.1)

In this formula, F and G are arbitrary smooth functions on R ²ⁿ and (q ₁, ..., q _n , p ₁ ..., p _n ) are linear coordinates on R ²ⁿ. He observed that if F and G are two first integrals of a mechanical system (defined on R ²ⁿ) then their Poisson bracket {F, G} is also a first integral. Notice that the Poisson bracket also allows one to describe the equations of motion in their most symmetric form

$$\begin{array}{*{20}{c}} {{{{\dot{q}}}_{i}} = \{ {{q}_{i}},H\} } \\ {{{{\dot{p}}}_{i}} = \{ {{p}_{i}},H\} } \\ \end{array} i = 1,...,n,$$

(3.2)

where H: R ²ⁿ →R is the Hamiltonian (the energy of the mechanical system, expressed in terms of position and momentum). Thirty years later, Jacobi explained (in [90]) Poisson’s observation by showing that the bracket (3.1) satisfies the identity

$$\left\{ {\left\{ {F,G} \right\},H} \right\} + \left\{ {\left\{ {G,H} \right\},F} \right\} + \left\{ {\left\{ {H,F} \right\},G} \right\} = 0$$

(3.3)

for all smooth functions F, G, H defined on R ²ⁿ. The above identity is now known as the Jacobi identity. To see how Poisson’s Theorem follows from the Jacobi identity it suffices to remark that K is a constant of the motion (3.2), precisely if K Poisson-commutes with H, i.e., {K, H} = 0, since

$$\dot K = \sum\limits_{i = 1}^n {\frac{{\partial K}}{{\partial {q_i}}}} \{ {q_i},H\} + \sum\limits_{i = 1}^n {\frac{{\partial K}}{{\partial {p_i}}}} \{ {p_i},H\} = \{ K,H\} .$$