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

One way to describe the dynamics of dissipative systems is via the Herglotz variational principle, in which the equation of motion is derived from a Lagrangian which depends on the action (i.e. the "Herglotz action"). In this presentation, I will discuss how the second functional derivative of the Herglotz action can be used to infer whether a given dissipative system is dynamically stable. I will illustrate these concepts by considering two examples: a harmonic oscillator with time-independent damping, and a harmonic oscillator with time-dependent damping (and time-dependent frequency).