The q-oscillator:: a Lagrangian description for variable damping

We propose a new parametric class of Lagrangians which incorporates time varying frictional effects to classical systems. This q-Lagrangian recovers the well-known Bateman's Lagrangian for the damped harmonic simple oscillator as a particular case. The equation of motion is integrated and the performance of this class of q-oscillators is compared with the harmonic oscillator under constant damping. A variety of qualitatively different dynamic behaviors can be observed when the free parameter q is continuously modified. An interesting property of the damped q-oscillator is that a stable harmonic oscillatory regime is always attained after a finite transient period. This behavior suggests a more realistic description for some physical systems where the energy is partially released in a finite time scale, at the end of which the system enters into a simple oscillatory regime (normal mode) as happens in earthquake dynamics. (C) 2000 Published by Elsevier Science B.V. All rights reserved.