  Unitary representations of the Galilei group are studied in phase space, in
order to describe classical and quantum systems. Conditions to write in general
form the generator of time translation and Lagrangians in phase space are then
established. In the classical case, Galilean invariance provides conditions for
writing the Liouville operator and Lagrangian for non-linear systems. We
analyze, as an example, a generalized kinetic equation where the collision term
is local and non-linear. The quantum counter-part of such unitary
representations are developed by using the Moyal (or star) product. Then a
non-linear Schr\"odinger equation in phase space is derived and analyzed. In
this case, an association with the Wigner formalism is established, which
provides a physical interpretation for the formalism.
