  We consider on a symplectic manifold M with Poisson bracket {,} an
Hamiltonian H with complete flow and a family Phi=(Phi_1,...,Phi_d) of
observables satisfying the condition {{Phi_j,H},H}=0 for each j. Under these
assumptions, we prove a new formula relating the time evolution of localisation
observables defined in terms of Phi to the variation of energy along classical
orbits. The correspondence between this formula and a formula established
recently in the framework of quantum mechanics is put into evidence.
  Among other examples, our theory applies to Stark Hamiltonians, homogeneous
Hamiltonians, purely kinetic Hamiltonians, the repulsive harmonic potential,
the simple pendulum, central force systems, the Poincare ball model, covering
manifolds, the wave equation, the nonlinear Schroedinger equation, the
Korteweg-de Vries equation and quantum Hamiltonians defined via expectation
values.
