Fourth-order dynamics of the damped harmonic oscillator

Abstract

It is shown that the classical damped harmonic oscillator belongs to the family of fourth-order Pais-Uhlenbeck oscillators. It follows that the solutions to the damped harmonic oscillator equation make the Pais-Uhlenbeck action stationary. Two systematic approaches are given for deriving the Pais-Uhlenbeck action from the damped harmonic oscillator equation, and it may be possible to use these methods to identify stationary action principles for other dissipative systems which do not conform to Hamilton's principle. It is also shown that for every damped harmonic oscillator $x$, there exists a two-parameter family of dual oscillators $y$ satisfying the Pais-Uhlenbeck equation. The damped harmonic oscillator and any of its duals can be interpreted as a system of two coupled oscillators with atypical spring stiffnesses (not necessarily positive and real-valued). For overdamped systems, the resulting coupled oscillators should be physically achievable and may have engineering applications. Finally, a new physical interpretation is given for the optimal damping ratio $\zeta=1/\sqrt{2}$ in control theory.