Abstract  The problem of the gauge variance or invariance of action functions in classical mechanics is discussed from a group and path-theoretic viewpoint. By using the elementary theory of the cohomology of groups, criteria are introduced which enable one to decide when action functions gauge variant under a kinematical group are equivalent to action functions invariant under the transformations of the group. The criteria are applied to action functions gauge variant under Lorentz and Galilei transformations, where we deduce that any action function gauge variant under the Lorentz group is equivalent to an action function invariant under Lorentz transformations, whilst action functions gauge variant under the Galilei group are not necessarily equivalent to Galilei-invariant action functions. It is also shown that any action function gauge variant in a more restricted fashion which we define in the text, is necessarily equivalent to a kinetic-energy action.