When action is not least

We examine the nature of the stationary character of the Hamilton action S for a space-time trajectory worldline x t of a single particle moving in one dimension with a general time-dependent potential energy function U x , t . We show that the action is a local minimum for sufficiently short worldlines for all potentials and for worldlines of any length in some potentials. For long enough worldlines in most time-independent potentials U x , the action is a saddle point, that is, a minimum with respect to some nearby alternative curves and a maximum with respect to others. The action is never a true maximum, that is, it is never greater along the actual worldline than along every nearby alternative curve. We illustrate these results for the harmonic oscillator, two different nonlinear oscillators, and a scattering system. We also briefly discuss two-dimensional examples, the Maupertuis action, and newer action principles. © 2007 American Association of Physics Teachers. DOI: 10.1119/1.2710480