Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 8 / 1 / i / Solution

A Poisson structure is a real bilinear operation on smooth functions on $M$,which is antisymmetric, obeys the Jacobi identity, and is a derivation in each argument. Explicitly,Antisymmetry supplies the corresponding Leibniz rule in the first argument. Equivalently it is a smooth Poisson bivector $\Pi$ with $\{f,g\}=\Pi (df,dg)$ and $[\Pi ,\Pi ]=0$ for the Schouten-Nijenhuis bracket. In local coordinates,The last equation is the coordinate Jacobi identity. Nondegeneracy is not part of the definition: a Poisson structure may have Casimir functions and symplectic leaves of different dimensions.

A Hamiltonian system consists of such a Poisson manifold, a Hamiltonian $H\in C^{\infty }(M)$, and the flow of its Hamiltonian vector field. Choose the conventionFor an observable with no explicit time dependence, $df/dt=\{f,H\}$. In particular $H$ is conserved because $\{H,H\}=0$. For the canonical Poisson bracket, this convention gives ${\dot{q}}_i=\partial H/\partial p_i$ and ${\dot{p}}_i=-\partial H/\partial q_i$.