A geometric generalization of classical mechanics and quantization

Abstract

A geometrization of classical mechanics is presented which may be considered as a realization of the Hertz picture of mechanics. The trajectories in the f-dimensional configuration space V _f of a classical mechanical system are obtained as the projections on V _f of the geodesics in an (f+1) dimensional Riemannian space V _f+1, with an appropriate metric, if the additional (f+1)th coordinate, taken to be an angle, is assumed to be "cyclic". When the additional (angular) coordinate is not cyclic we obtain what may be regarded as a generalization of classical mechanics in a geometrized form. This defines new motions in the neighbourhood of the classical motions. It has been shown that, when the angular coordinate is "quasi-cyclic", these new motions can be used to describe events in the quantum domain with appropriate periodicity conditions on the geodesics in V _{f+ 1}.