When action is not least for systems with action-dependent Lagrangians

TL;DR

This paper extends the variational principle of Herglotz to analyze the stability of non-conservative and dissipative systems through the second variation of an action-dependent action, with applications to harmonic oscillators.

Contribution

It generalizes the second variation analysis from conservative to action-dependent systems, enabling stability assessment of non-conservative dynamics.

Findings

Second variation of action-dependent action can determine system stability.

Method applied successfully to harmonic oscillator models.

Provides a framework for analyzing dissipative systems' dynamics.

Abstract

The dynamics of some non-conservative and dissipative systems can be derived by calculating the first variation of an action-dependent action, according to the variational principle of Herglotz. This is directly analogous to the variational principle of Hamilton commonly used to derive the dynamics of conservative systems. In a similar fashion, just as the second variation of a conservative system's action can be used to infer whether that system's possible trajectories are dynamically stable, so too can the second variation of the action-dependent action be used to infer whether the possible trajectories of non-conservative and dissipative systems are dynamically stable. In this paper I show, generalizing earlier analyses of the second variation of the action for conservative systems, how to calculate the second variation of the action-dependent action and how to apply it to two physically important systems: a time-independent harmonic oscillator and a time-dependent harmonic oscillator.