Invariance and Conservation Laws in Classical Mechanics

Abstract

It is commonly supposed that, in classical mechanics, invariance of a physical system to coordinate translation implies conservation of linear momentum. ``Invariance'' may be defined in a number of ways. If it is defined to mean invariance of the equation of motion, it is shown that invariance of this equation with respect to coordinate translation does not imply conservation of linear momentum. The effects of scale transformation and coordinate inversion invariance are also investigated. Both the Lagrangian approach and Newton's (second) law approach are considered. It is shown that each of the above invariances implies a condition on the equation of motion, while a combination of these and time-inversion invariance is needed to obtain ordinary momentum conservation.