Classical Noether's theory with application to the linearly damped particle

TL;DR

This paper revisits Noether's theory in classical mechanics, applying it to a damped particle system, analyzing symmetries, and exploring extensions like non-local constants, with a focus on educational clarity and physical interpretation.

Contribution

It provides a modern, detailed analysis of Noether's symmetries in dissipative systems, including new methods for transforming Lagrangians and extending symmetry concepts.

Findings

Identified all potentials with time-independent symmetries for damped particles.

Developed a symmetry-based transformation to convert Lagrangians into autonomous forms.

Connected Noether's theory with non-local constants and on-flow symmetries.

Abstract

This paper provides a modern presentation of Noether's theory in the realm of classical dynamics, with application to the problem of a particle submitted to both a potential and a linear dissipation. After a review of the close relationships between Noether symmetries and first integrals, we investigate the variational point symmetries of the Lagrangian introduced by Bateman, Caldirola and Kanai. This analysis leads to the determination of all the time-independent potentials allowing such symmetries, in the one-dimensional and the radial cases. Then we develop a symmetry-based transformation of Lagrangians into autonomous others, and apply it to our problem. To be complete, we enlarge the study to Lie point symmetries which we associate logically to Noether ones. Finally, we succinctly address the issue of a `weakened' Noether's theory, in connection with on-flows symmetries and non-local constant of motions, for it has a direct physical interpretation in our specific problem. Since the Lagrangian we use gives rise to simple calculations, we hope that this work will be of didactic interest to graduate students, and give teaching material as well as food for thought for physicists regarding Noether's theory and the recent developments around the idea of symmetry in classical mechanics.