ABSTRACT
In this chapter, the authors consider the concept of an adiabatic invariant (which has been little studied by mathematicians in spite of its importance and interest). They investigate the variation of an adiabatic invariant in the course of an infinite interval of time for a small periodic variation of the parameters of an oscillating system with one degree of freedom. A qualitative investigation of the behaviour of solutions of a system of ordinary differential equations usually begins with the finding of individual, particularly simple solutions: positions of equilibrium and periodic trajectories. After that the distribution of integral curves in the neighbourhood of these solutions is investigated which sometimes leads to important conclusions regarding the behaviour of the solutions as a whole. The “ergodic hypothesis” of statistical mechanics is concerned with the idea that in a dynamical system of “general form” motion on a surface of constant energy has properties of ergodicity and intermixing.
