HAMILTONIAN-MECHANICS OF THE DAMPED HARMONIC-OSCILLATOR

It has long been known that an exponential transformation can eliminate the second highest derivative in an ordinary linear differential equation. For example, Lamb analyzes the (non-dimensionalized) harmonic oscillator with viscous damping x + γx + x = 0 by introducing the 'reducing' transformation. It is easily verified that the canonical equations x = &partHx/&partPx, Px = - &partHx/&partx are equivalent to the first equation. Authors have thus obtained a canonical transformation of the damped harmonic oscillator into an undamped oscillator. Furthermore, they have shown that any two oscillators with different damping constants are canonically equivalent, including the case when one damping constant vanishes. In addition, they make several interesting observations which are included in the article.