When is Hamilton's Principle an Extremum Principle

Abstract

For a dynamic system, the action / for motion from time f0 to time t can be written as I = ^(T— V)dt, where T is the kinetic energy and V is the potential energy. In many references to Hamilton's principle, particularly in the engineering literature, it is stated or implied that the true motion of the system will give the action an extreme value, usually stated to be a minimum value. In this Paper it is first demonstrated by very simple examples that Hamilton's principle is not, in general, an extremum principle. Then a relatively elementary and direct proof, which does not require sufficiency theory from the calculus of variations, is presented which shows that for certain discrete linear systems the action is always minimized over short time intervals; and a precise characterization is given for the maximum length of the time interval over which the action is guaranteed to be minimized by the true solution. Finally, it is shown that for continuous systems the action is never minimized by the true solution.