Invariances of approximately relativistic Hamiltonians and the center-of-mass theorem

Abstract

In an earlier paper we considered a class of Lagrangians for directly interacting particles, arising from a slow-motion approximation in various special- and general-relativistic field theories. It was shown that if the Lagrangian is invariant under time and space translations this implies invariance under an additional three-parameter set of infinitesimal transformations, which leads directly to the center-of-mass theorem. This result is rederived here in a Hamiltonian formalism, in which these infinitesimal transformations are shown to be generators of a Lie symmetry group in phase space. Then we consider the problem of the most general form possible of a canonical post-Newtonian theory that is a realization of the Lie algebra of the Poincar\'e group to order ${c}^{\ensuremath{-}2}$ and that arises from a theory of the usual Newtonian type with two-body interactions. It is found that in such a theory the world-line condition is satisfied to order ${c}^{\ensuremath{-}2}$. This canonical theory encompasses all the approximately relativistic interactions, found recently by Woodcock and Havas, which follow from a Fokker-type special-relativistic variational principle for particles with direct two-body interactions. The relation of our work to various other approaches to approximately relativistic theories of interacting particles is discussed.