Classical Mechanics of Rotational–Translational and Other Energy Transfer. I. A Hamilton–Jacobi (Action-Angle) Treatment

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Abstract
The classical-mechanical equations of motion describing collisional energy transfer are converted to a form involving constants of the elastic collision, using a Hamilton–Jacobi formalism. These constants frequently vary only slowly during an inelastic collision. An approximate version of the transformed equations, related to a variation of constants procedure, is next introduced. The collision of an atom with a rigid linear diatomic molecule is considered in some detail. Several desirable features of the approximation are that the change in rotational angular momentum is obtained directly, all initial orientations of particles and angular momenta occur outside the integrals, an approximate error estimate can be made, results can be calculated relatively quickly, and further insight is obtained into the energy-transfer process. Because of the close relationship of Hamilton–Jacobi and Schrödinger formalisms, a comparison of exact and approximate classical results is also expected to provide estimates of range of validity of some commonly used approximations in the quantum case.