  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.
