When action is not least

Abstract

We examine the nature of the stationary character of the Hamilton action S for a space-time trajectory ͑worldline͒ x͑t͒ of a single particle moving in one dimension with a general time-dependent potential energy function U͑x , t͒. 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 U͑x͒, 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.

About non-maximality of the action functional

“…In 2007, C. Gray and E. Taylor 8 presented a discussion about non-maximality of stationary action and showed that the solution of the equation of the simple harmonic oscillator is a saddle point of the action by assuming that the time in the Lagrangian integral is greater than the semi-period of the oscillator, which is physically relevant to allow oscillations in this range. This is also contained in Chapter 6 of the book of David Morin.…”

“…8, which cite Refs. 9, 11, and 12) of a result which states that the action can never be maximum if the potential of the Lagrangian is velocity-independent, leaving for this potential, only the case of minimum or saddle.…”

“…10 and so we realize a "rigorous proof" of non-maximality of the action functional for potentials V (x, t) velocity-independent (A study to determine whether the non-maximality of the action of a given system is minimum or saddle was made by Gray and Taylor in Ref. 8 and involves the lenghts of the world lines and, in some cases, higher order derivatives. ).…”

Staticization and Associated Hamilton-Jacobi and Riccati Equations

“…This latter viewpoint appears particularly useful in some applications in modern physics, including gravitational systems where relativistic effects are non-negligible, and systems in the quantum domain (cf. [6,7,8,13]). Our interests are more pedestrian; the stationary-action formulation has recently been found to be quite useful for generation of fundamental solutions to two-point boundary-value problems (TPBVPs) for conservative dynamical systems.…”

An Evolutionary View on Function-Based Stability

“…To explain the dynamics behind it would lead too far away from the presentation's focus, but reading QED and [9] is strongly recommended to deepen understanding of the far reaching influence of the PSA. For a deeper understanding of the quantummechanical basis of action, reading [10] is recommended.…”