Geometric conditions for the positive definiteness of the second variation in one-dimensional problems

TL;DR

This paper introduces a geometric approach to determine the positive definiteness of the second variation in one-dimensional problems, simplifying stability analysis of stationary solutions through phase-plane trajectory analysis.

Contribution

It provides a novel geometric method to assess the positive definiteness of the second variation, offering a straightforward proof and application to mechanical stability problems.

Findings

Phase-plane geometry determines minimality of solutions.

The method simplifies stability analysis for mechanical systems.

Application to a hanging mass problem demonstrates practical utility.

Abstract

Given a functional for a one-dimensional physical system, a classical problem is to minimize it by finding stationary solutions and then checking the positive definiteness of the second variation. Establishing the positive definiteness is, in general, analytically untractable. However, we show here that a global geometric analysis of the phase-plane trajectories associated with the stationary solutions leads to generic conditions for minimality. These results provide a straightforward and direct proof of positive definiteness, or lack thereof, in many important cases. In particular, when applied to mechanical systems, the stability or instability of entire classes of solutions can be obtained effortlessly from their geometry in phase-plane, as illustrated on a problem of a mass hanging from an elastic rod with intrinsic curvature.