A formula relating sojourn times to the time of arrival in Hamiltonian dynamics
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2012
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Abstract
We consider on a manifold M equipped with a Poisson bracket {., .} a Hamiltonian H with complete flow and a family Phi = (Phi(1), ... , Phi(d)) of abstract position observables satisfying the condition {{Phi(j), H}, H} = 0 for each j. Under these assumptions, we prove a new formula relating sojourn times in dilated regions defined in terms of Phi to the time of arrival of 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, central force systems, the Poincare ball model, the wave equation, the nonlinear Schrodinger equation, the Korteweg-de Vries equation and quantum Hamiltonians defined via expectation values.