  Certain dissipative systems, such as Caldirola and Kannai's damped simple
harmonic oscillator, may be modelled by time-dependent Lagrangian and hence
time dependent Hamiltonian systems with $n$ degrees of freedom. In this paper
we treat these systems, their projective and conformal symmetries as well as
their quantisation from the point of view of the Eisenhart lift to a Bargmann
spacetime in $n+2$ dimensions, equipped with its covariantly constant null
Killing vector field. Reparametrization of the time variable corresponds to
conformal rescalings of the Bargmann metric. We show how the Arnold map lifts
to Bargmann spacetime. We contrast the greater generality of the
Caldirola-Kannai approach with that of Arnold and Bateman. At the level of
quantum mechanics, we are able to show how the relevant Schr\"odinger equation
emerges naturally using the techniques of quantum field theory in curved
spacetimes, since a covariantly constant null Killing vector field gives rise
to well defined one particle Hilbert space. Time-dependent Lagrangians arise
naturally also in cosmology and give rise to the phenomenon of Hubble friction.
We provide an account of this for Friedmann-Lemaitre and Bianchi cosmologies
and how it fits in with our previous discussion in the non-relativistic limit.
