Some Insights into Jacobi's Form of Least Action Principle

Abstract

Jacobi's form of least action principle is generally known as a principle of stationary action. The principle is studied, in the view of calculus of variations, for the minimality and the existence of trajectory that connects two prescribed configurations. It is found, by utilizing a finitely compact topology on the configuration space, that every pair of configurations can be connected by a minimal curve. Therefore it is a principle of minimum action if the corner condition is allowed. If the set of rest configurations (zero kinetic energy) is empty, then the minimal curve is a minimal trajectory, implying that Fermat's Principle for the geometrical optics is a minimum principle because the speed of light does not vanish. If that set is not empty, then a minimal curve may either be a minimal trajectory or consist of minimal trajectories and curves Cl lying on the surface of rest configurations. Each one of curves Cl forms a corner. A minimal trajectory satisfies the Euler–Lagrange equation and has the property that the action is minimum among all curves lying in configuration space.