A formula relating localisation observables to the variation of energy in Hamiltonian dynamics

TL;DR

This paper derives a new classical formula linking localisation observables to energy variation in Hamiltonian systems, with applications to various physical models and quantum analogs.

Contribution

It introduces a novel formula connecting localisation observables to energy change, applicable to a wide range of classical and quantum Hamiltonian systems.

Findings

Formula relates localisation observables to energy variation.

Applicable to diverse Hamiltonian systems including quantum and classical models.

Establishes a link between classical and quantum formulations.

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 Korteweg-de Vries equation and quantum Hamiltonians defined via expectation values.