Noether symmetries and conservation laws in the classical problem of a particle with linear damping

Raphaël Leone, Thierry Gourieux

Published 2014-12-23Version 1

This paper is devoted to the search for conservation laws in the unidimensional problem of a particle submitted to both a potential and a linear dissipation. After a review of Noether's theory and its relation to first integrals we analyse the variational point symmetries of the Lagrangian introduced by Bateman, Caldirola and Kanai. We find all the time-independent potentials allowing such symmetries. 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 show that any first integral of the problem amounts to a local expression of the Bateman-Caldirola-Kanai action in the phase space, along the solution curves.