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

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.

Accepted for publication in the Journal of Mathematical Physics

References in corpus (6)

• Array Programming with NumPy
• From an Action Principle for Action-dependent Lagrangians toward non-conservative Gravity: accelerating Universe without dark energy
• Invariant vacuum
• The action principle for dissipative systems
• Generalized nonconservative gravitational field equations from Herglotz action principle
• Application of Herglotz's Variational Principle to Electromagnetic Systems with Dissipation