  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.
