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

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.

Automated summary

The dynamics of some non-conservative and dissipative systems can be derived using the first variation of an action-dependent action based on Herglotz's variational principle. Similar to the Hamiltonian variational principle for conservative systems, the second variation of the action-dependent action can determine the dynamic stability of possible trajectories for non-conservative and dissipative systems. This paper generalizes previous analyses and demonstrates how to calculate the second variation of the action-dependent action, applying it to both the time-independent and time-dependent harmonic oscillators.