The principle of stationary action in the calculus of variations

TL;DR

This paper reviews mathematical techniques from non-linear analysis to rigorously determine when critical paths of the action functional in physics are minima, extending analysis beyond traditional physical reasoning to a broader class of models.

Contribution

It develops precise mathematical conditions for critical paths to be minima in various models, enabling systematic analysis of non-trivial physical models.

Findings

Established criteria for minima of the action functional

Applied techniques to modern physical models

Extended analysis beyond traditional physical reasoning

Abstract

We review some techniques from non-linear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Previous attempts to analyse when these are minima ex- ist, but mainly based on physical reasoning and only for a restricted class of models. Our main intention in this regard is to develop precise mathematical conditions for critical paths to be minimum solutions in a variety of situations. Our claim is that, with a few techniques, a systematic analysis (including the domain for which critical points are genuine minima) of non-trivial models is possible. We present specific models arising in modern physical theories in order to make clear the ideas here exposed.