 114 Antoine Gournay, Rafael Tiedra de Aldecoa
 A formula relating localisation observables to the
variation of energy in Hamiltonian dynamics
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Jan 8, 11

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Abstract. 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 Kortewegde Vries equation and
quantum Hamiltonians defined via expectation values.
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