Numerical study of particle motion in two waves

Abstract

We are studying the dynamical system described by the Hamiltonian H = H₀ + ɛV, where

$$H_0 = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/\kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} p^2 - \cos x, V = - cos (\lambda x - \Omega t).$$

We have encountered this system in a number of problems of practical importance. In addition, the system has intrinsic interest for the theory of adiabaticity and stochasticity. The invariant action J of the unperturbed Hamiltonian H₀ is subject to strong modification or destruction because of the perturbation ɛV. Absence of an invariant (i.e., stochasticity) occurs in a phase space region whose size and shape vary with the three parameters ɛ, λ, Ω. Previous studies have varied the amplitude of a perturbation (our ɛ); we emphasize here the strong dependences on the space (λ) and time (Ω) scales of the perturbation. Our results show that a perturbation is most effective at causing stochastic motion if its space and time scales are comparable (λ∼1, Ω∼1) to those in the unperturbed Hamiltonian H₀.