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

Abstract

Abstract. The classical Hamiltonian system of time-dependent harmonic oscillator driven by the arbitrary external time-dependent force is considered. Exact analytical solution of the corresponding equations of motion is constructed in the framework of the technique (Robnik M, Romanovski V G, J. Phys. A: Math. Gen. 33 (2000) 5093) 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 characteristic 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.

Statistical properties of 1D time-dependent Hamiltonian systems: from the adiabatic limit to the parametrically kicked systems

“…Describing P (E 1 ) in general is a difficult problem, but for the 1D general time-dependent harmonic oscillator, the problem can be solved [8][9][10]12]. We do not consider the external forcing any further, but the details can be found in [11]. The system is described by the Newton equation q + ω 2 (t)q = 0, (1) and we work out rigorously P (E 1 ).…”

“…where we have denoted δ = (α − β)/2. It follows from ( 9) and ( 10) that we can write (11) also in the form…”

“…But aperiodic time variation of the system (control) parameters is also interesting and important, especially in the case of sufficiently slow variation, in the so-called adiabatic limit, even in the 1D systems. The questions of the energy evolution of a microcanonical ensemble of initial conditions and the preservation of the adiabatic invariants are very closely related, and they have been addressed for the entirely general 1D time-dependent linear oscillator in a series of papers by Robnik and Romanovski [8][9][10][11][12]. A remarkable theorem has been found there: in full generality, the adiabatic invariant at the mean energy of an initial microcanonical ensemble never decreases.…”

WKB approach applied to 1D time-dependent nonlinear Hamiltonian oscillators

“…Recently, Robnik and Romanovski managed to solve explicitly and rigorously the discrete WKB recurrence formula to all orders [2] and then applied the method to solvable 1D potentials in the context of quantum mechanics to all orders, showing that in this way we can get exact energy spectra for all solvable 1D potentials with discrete energy spectrum [3][4][5][6][7][8]. Later on, in a series of papers, they successfully applied the method to all orders of the time-dependent linear oscillator with arbitrary time dependence of the oscillator frequency [9][10][11][12][13][14] and discussed in detail the question of the preservation of the adiabatic invariants and its relationship with the statistical properties of such systems, such as the evolution of the energy distribution, starting from an initial microcanonical ensemble of initial conditions. Time-dependent Hamiltonian systems are very interesting and important dynamical models, where many important questions about their dynamical statistical behavior can be studied [15][16][17][18][19].…”

“…But non-periodic time variation of the system (control) parameters is also interesting and important, especially in the case of sufficiently slow variation, in the so-called adiabatic limit, even in the 1D systems. The problems of the energy evolution of a microcanonical ensemble of initial conditions and the preservation of the adiabatic invariants are very closely related, and they have been studied for the entirely general 1D time-dependent linear oscillator in the above-mentioned series of papers by Robnik and Romanovski [9][10][11][12][13][14]. An interesting and important theorem has been found and proved there; in full generality, the adiabatic invariant at the mean final energy of an initially microcanonical ensemble never decreases.…”