WHAT IS POISSON GEOMETRY?

In Classical Mechanics one learns how to describe a mechaninal system with n degrees of freedom evolving with time. Briefly, the state of the system at time t is described by a point (q(t), p(t)) in phase space R2n. Here the (q1(t),..., qn(t)) are the configuration coordinates and the (p1(t),..., pn(t)) are the momentum coordinates of the system. The evolution of the system in time is determined by a function h: R2n → R, called the hamiltonian: if (q(0), p(0)) is the initial state of the system, then the state at time t is obtained by solving Hamilton’s equations: (1) q̇i = ∂H∂pi, ṗi = −∂H∂qi, (i = 1,..., n). This description of mechanics is the departing point for Poisson geometry. First, one starts by defining a new product {f1, f2} between any two smooth functions f1 and f2, called the Poisson bracket, by setting: (2) {f1, f2}:= n∑ i=1 ∂f