Energy evolution in the time-dependent harmonic oscillator with arbitrary external forcing

Abstract

The classical Hamiltonian system of time-dependent harmonic oscillator driven by an arbi- trary external time-dependent force is considered. Exact analytical solution of the corresponding equations of motion is constructed in the framework of the technique based on WKB approach. Energy evolution for the ensemble of uniformly distributed w.r.t, the canonical angle initial conditions on the initial invariant torus is studied. Exact expressions for the energy moments of arbitrary order taken at arbitrary time moment are analytically derived. Corresponding character- istic function is analytically constructed in the form of infinite series and numerically evaluated for certain values of the system parameters. Energy distribution function is numerically obtained in some particular cases. In the limit of small initial ensemble's energy the relevant formula for the energy distribution function is analytically derived. Keywords: classical dynamics, Hamiltonian systems, linear oscillator, energy distribution func- tion, WKB approach.