3 - Hamilton's Principle

Abstract

This chapter is designed to express the equations in the form of stationary values of a time integral. The notion of zero variation of a quantity was seen in the technique of virtual work and extended to dynamics by means of D'Alembert's principle. It has long been considered that nature works so as to minimize some quantity often called action. One of the first statements was made by Maupertuis in 1744. The most frequently used form is that devised by Sir William Rowan Hamilton around 1834. Hamilton's principle could be considered to be a fundamental statement of mechanics, especially as it has extensive applications in other areas of physics. This chapter develops the principle directly from Newtonian laws. For the case with conservative forces the principle states that the time integral of the Lagrangian is stationary with respect to variations in the path in configuration space. That is, the correct displacement–time relationships give a minimum or maximum value of the integral. One of the areas in which Hamilton's principle is useful is that of continuous media where the number of degrees of freedom is infinite. In particular it is helpful in complex problems for which approximate solutions are sought, because approximations in energy terms are often easier to see than they are in compatibility requirements.