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

## TL;DR

This paper shows that simple and damped harmonic oscillators share an underlying Lie algebra structure, enabling the treatment of dissipative systems within the orbit method and providing new insights into their symmetries.

## Contribution

It introduces a novel approach to analyze dissipative systems using the orbit method and explores the coadjoint orbits of the (1+1) Newton-Hooke group.

## Key findings

- Simple and damped oscillators are indistinguishable at the Lie algebra level.
- Coadjoint orbits of the (1+1) Newton-Hooke group are characterized.
- Physical interpretation involves a realization of the Lie algebra on phase space.

## Abstract

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.