Poisson Bracket - Hamilton's Equations of Motion

Hamilton's Equations of Motion

The Hamilton's equations of motion have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that f(p,q,t) is a function on the manifold. Then one has

Further, by taking p = p(t) and q = q(t) to be solutions to Hamilton's equations

and

one may write

Thus, the time evolution of a function f on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e. canonical transformations, area-preserving diffeomorphisms), with the time t being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that any time t in the solution to Hamilton's equations, q(t)=exp(-t{H,•}) q(0), p(t)=exp(-t{H,•}) p(0), can serve as the bracket coordinates. Poisson brackets are canonical invariants.

Dropping the coordinates, one has

The operator in the convective part of the derivative, is sometimes referred to as the Liouvillian (see Liouville's theorem (Hamiltonian)).