Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian

Abstract

If an integrable classical Hamiltonian H describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables I of the motion are conserved. Here the fate of the angle variables is analysed. Because of the unavoidable arbitrariness in their definition, angle variables belonging to distinct initial and final Hamiltonians cannot generally be compared. However, they can be compared if the Hamiltonian is taken on a closed excursion in parameter space so that initial and final Hamiltonians are the same. The result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega =dH/dI, but differs from it by a definite extra angle which depends only on the circuit in parameter space, not on the duration of the process. The 2-form which describes this angle variable holonomy is calculated.