An Adiabatic Theorem for Motions Which Exhibit Invariant Phase Curves

TL;DRAbstract

We consider a particle with one degree of freedom moving according to a Hamiltonian function H(p, q, λ, t), depending on a parameter λ, and which may also depend on the time. We assume that for each fixed value of A of interest, the motion is stable and exhibits invariant phase curves, that is, curves which are either continuously or periodically transformed into themselves by the transformation generated by the Hamiltonian equations of motion. This implies that there is a constant of the motion α(p, q, λ, t), which is either independent of t or periodic in t, the invariant curves being given by the equation α(p, q, λ, t) = α = a constant. When the Hamiltonian is independent of t, the invariant curves are simply the closed orbits on the phase plane. When the Hamiltonian is periodic in t, there is evidence, at least in many cases, for the existence of invariants of the form (1) with the same period in t. We will further assume that there is only one set of invariant curves, that is, tha

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