Statistical properties of the energy in time-dependent homogeneous power law potentials

We study classical 1D Hamilton systems with a homogeneous power law potential and their statistical behavior, assuming a microcanonical distribution of the initial conditions and describing its change under a monotonically increasing time-dependent function a(t) (the prefactor of the potential). Using the nonlinear Wentzel-Kramers-Brillouin-like method of Papamikos and Robnik 2012 J. Phys. A: Math. Theor. 44 315102 and following a previous work by Papamikos and Robnik 2011 J. Phys. A: Math. Theor. 45 015206, we specifically analyze the mean energy, the variance and the adiabatic invariant (action) of the systems for large time t -> infinity and we show that the mean energy and variance increase as powers of a(t), while the action oscillates and finally remains constant. By means of a number of detailed case studies, we show that the theoretical prediction is excellent, which demonstrates the usefulness of the method in such applications.