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Statistical properties of 1D time-dependent Hamiltonian systems: from the adiabatic limit to the parametrically kicked systems
Papamikos, Georgios ; Robnik, Marko, 1954-
We consider 1D time-dependent Hamiltonian systems and their statistical properties, namely the time evolution of microcanonical distributions, whose properties are very closely related to the ...existence and preservation of the adiabatic invariants. We review the elements of the recent developments by Robnik and Romanovski (during 2006-2008) for the entirely general 1D time-dependent linear oscillator and try to generalize the results to the 1D nonlinear Hamilton oscillators, in particular the power-law potentials like e.g. the quartic oscillator. Furthermore, we consider the limit opposite to the adiabatic limit, namely parametrically kicked 1D Hamiltonian systems. Evenfor the linear oscillator, interesting properties are revealed: an initial kick disperses the microcanonical distribution to a spread one, but an anti-kick at an appropriate moment of time can annihilate it, kicking it back to the microcanonical distribution. The case of periodic parametric kicking is also interesting. Finally, we propose that in the parametric kicking of a general 1D Hamilton system, the average value of the adiabatic invariant always increases, which we prove for the power-law potentials. We find that the approximation of kicking is good for quite long times of the parameter variation, up to the order of not much less than one period of the oscillator. We also look at the behavior of the quartic oscillator for the case of the kick and anti-kick, and also the periodic kicking.
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