(1+1) Newton-Hooke Group for the Simple and Damped Harmonic Oscillator

It is demonstrated that, in the framework of the orbit method, a simple and damped harmonic oscillators are indistinguishable at the level of an abstract Lie algebra. This opens a possibility for treating the dissipative systems within the orbit method. In depth analysis of the coadjoint orbits of the $(1+1)$ dimensional Newton-Hooke group are presented. Further, it is argued that the physical interpretation is carried by a specific realisation of the Lie algebra of smooth functions on a phase space rather than by an abstract Lie algebra.