About non-maximality of the action functional

Abstract

In this work first we review some cases where the action exhibits a minimal or a saddle-point criticality for velocity-independent potentials (V(x, t)) and maximum when the potential is velocity-dependent (V(x,ẋ,t)). In the following we will use the functional (“directional”) derivative of second order to present a mathematically rigorous proof of the non-maximality of the classical functional action for potentials V(x, t) velocity-independent.