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.
