Some insights into Jacobi's form of least action principle

Jacobi's form of least action principle is generally known as a principle o f stationary action. The principle is studied, in the view of calculus of v ariations, for the minimality and the existence of trajectory that connects two prescribed configurations. It is found, by utilizing a finitely compac t topology on the configuration space, that every pair of configurations ca n be connected by a minimal curve. Therefore it is a principle of minimum a ction if the corner condition is allowed. If the set of rest configurations (zero kinetic energy) is empty, then the minimal curve is a minimal trajec tory, implying that Fermat's Principle for the geometrical optics is a mini mum principle because the speed of light does not vanish, If that set is no t empty, then a minimal curve may either be a minimal trajectory or consist of minimal trajectories and curves C-l lying on the surface of rest config urations. Each one of curves C-l forms a corner. A minimal trajectory satis fies the Euler-Lagrange equation and has the property that the action is mi nimum among all curves lying in configuration space. (C) 2001 Academic Pres s.