Kinematics of trajectories in classical mechanics

Abstract

In this paper, we show how the study of kinematics of a family of trajectories of a classical mechanical system may be unified within the framework of analysis of geodesic flows in Riemannian geometry and Relativity. After setting up the general formalism, we explore it through studies on various one and two dimensional systems. Quantities like expansion, shear and rotation (ESR), which are more familiar to the relativist, now re-appear while studying such families of trajectories in configuration space, in very simple mechanical systems. The convergence/divergence of a family of trajectories during the course of time evolution, the shear and twist of the area enclosing the family, and the focusing/defocusing of the trajectories within a finite time are investigated analytically for these systems. The understanding of the configuration space developed through such investigations is elaborated upon, and possible future avenues are pointed out.