Variational mechanics in one and two dimensions

Abstract

We develop heuristic derivations of two alternative principles of least action. A particle moving in one dimension can reverse direction at will if energy conservation is the only criterion. Such arbitrary changes in the direction of motion are eliminated by demanding that the Maupertuis–Euler abbreviated action, equal to the area under the momentum versus position curve in phase space, has the smallest possible value consistent with conservation of energy. Minimizing the abbreviated action predicts particle trajectories in two and three dimensions and leads to the more powerful Hamilton principle of least action, which not only generates conservation of energy, but also predicts motion even when the potential energy changes with time. Introducing action early in the physics program requires modernizing the current obscure and confusing terminology of variational mechanics.