Effective Dynamics for a Mechanical Particle Coupled to a Wave Field

Abstract

We consider a particle coupled to a scalar wave field and subject to the slowly varying potential V ("q) with small ". We prove that if the initial state is close, order " 2 , to a soliton (=dressed particle), then the solution stays forever close to the soliton manifold. This estimate implies that over a time span of order" 1 the radiation losses are negligible and that the motion of the particle is governed by the effective Hamiltonian Heff (q;P )= E(P )+ V ("q) with energy-momentum relationE(P ).