  We study 1D Hamilton systems with homogeneous power law potential and their
statistical behaviour, assuming the microcanonical distribution of the initial
conditions and describing its change under monotonically increasing
time-dependent function $a(t)$ (prefactor of the potential). Using the
nonlinear WKB-like method by Papamikos and Robnik {\em J. Phys. A: Math. Theor.
{\bf 44} (2012) 315102} and following a previous work by Papamikos G and Robnik
M {\em J. Phys. A: Math. Theor. {\bf 45} (2011) 015206} we specifically analyze
the mean energy, the variance and the adiabatic invariant (action) of the
systems for large time $t\rightarrow\infty$ 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.
