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Recently there has been renewed interest in adiabatic invariants of simply-periodic classical systems subject to perturbation by slow variation of parameters. In several interesting cases it has been shown that if the system varies slowly from one steady state to a different steady state, the appropriate adiabatic invariant is constant to an arbitrarily high order in a slowness parameter.It is shown here how these and similar results may be derived by systematic use of a technique of perturbation theory of classical Hamiltonian systems. The method is essentially iteration of the transformation to action and angle variables.

In molecular encounters, the total energy of the system may not be sufficient to enable any but the ground-state electronic surface to determine nuclear motions adiabatically. When this happens, non-adiabatic transitions are energetically forbidden. It is shown that for such collisions the phase shifts computed in the adiabatic approximation are lower bounds to the exact phase shifts. The proof derives from a study of the non-local potential which arises when the exact collision problem is transformed so as to involve only nuclear motion explicitly. Use is made of a well-known relation between magnitudes of effective potentials and of phase shifts associated with two respective hamiltonians [1, 2 (b)]. This result is related to a theorem by Hahn et al. [2 (a]. The similarities and differences between the two are discussed.