TIME-TRANSLATION INVARIANCE FOR CERTAIN DISSIPATIVE CLASSICAL SYSTEMS

Time-translation invariance of the equation of motion for an arbitrary one-dimensional classical system has been shown to generate a conservation law. For conservative systems, this conserved quantity is shown here to be the usual energy. For the linearly damped harmonic oscillator, the explicit form of this constant is found, as well as for quadratically damped systems. In both cases, as the damping approaches zero, the conserved quantity is shown to approach a function of the usual energy. For both types of dissipation, Lagrangians are given which yield the equations of motion. © 1968, American Association of Physics Teachers. All rights reserved.