Lagrangian formulation and first integrals of piecewise linear dissipative systems

The Lagrangian description of the motion of a damped oscillator is used as a starting point for deducing the canonical formulation and studying the existence of first integrals according to Noether's generalized theorem. It is shown that first integrals exist only if the restoring force is linear. Finally, results are applied to the study of some important linear or piecewise-linear cases such as the Reid oscillator and the oscillator with bilinear hysteresis.