  Closed systems in Newtonian mechanics obey the principle of Galilean
relativity. However, the usual Lagrangian for Newtonian mechanics, formed from
the difference of kinetic and potential energies, is not invariant under the
full group of Galilean transformations. In quantum mechanics Galilean boosts
require a non-trivial transformation rule for the wave function and a
concomitant "projective representation" of the Galilean symmetry group. Using
Feynman's path integral formalism this latter result can be shown to be
equivalent to the non-invariance of the Lagrangian. Thus, using path integral
methods, the representation of certain symmetry groups in quantum mechanics can
be simply understood in terms of the transformation properties of the classical
Lagrangian and conversely. The main results reported here should be accessible
to students and teachers of physics -- particularly classical mechanics,
quantum mechanics, and mathematical physics -- at the advanced undergraduate
and beginning graduate levels, providing a useful exposition for those wanting
to explore topics such as the path integral formalism for quantum mechanics,
relativity principles, Lagrangian mechanics, and representations of symmetries
in classical and quantum mechanics.
