Invariance and Conservation Laws for a Classical Relativistic Particle

Abstract

The general equation of motion of a classical relativistic particle, moving in one dimension, is examined to determine what results are implied by the invariance of this equation to a number of well-known space-time transformations, i.e., the space and time translations, scale changes and inversions, and the Lorentz transformations. While the invariance of the equation to a translation or Lorentz transformation alone generates a conservation law, invariance to the others does not. Conditions on the Lagrangian describing the system for invariance to these transformations are also given. While invariant Lagrangians yield equations of motion with the same invariance, Lagrangians exist that are not invariant to certain transformations but that produce invariant equations of motion. For the Lorentz transformations, the invariance of ∫Ldt is also examined. Further, it is shown that invariance of the equation of motion to Lorentz transformations provides a definition of the relativistic linear momentum that is independent of the Lagrangian or any assumed conservation law.