The Use of the Linear Combination of Physical Solutions in the Study of the Nature of the Stationary Point of Action

Abstract

Hamilton’s principle states that among all curves between two end points, the path actually followed by a physical system will always be the one that gives an stationary value (minimum, maximum or a saddle points) to the Hamilton’s action (an integral over time of the difference between the kinetic and the potential energy, taken between the initial time and the final time of the development of the system). It is common to use a mathematical tool called Calculus of Variations to study the Hamilton’s principle. The Calculus of Variations deals with functionals (functions of functions) and is a more general and more complex version of the usual calculus that we learn in university. In this paper we present an alternative, and simpler, approach to the study of the Hamilton’s principle. We study the nature of the action’s stationary space-time trajectory of three physical systems: free-particle, vertical launch and harmonic oscillator, using as virtual motion a linear combination of the physical solution of this three systems. We find evidences that the physical solution of the free-particle problem leads to a minimum on its action. The same results occours on the vertical launch problem. The physical solution of the harmonic oscillator leads to a minimum or a saddle point on its action deppending on the time interval of the development of the system.