Invariances of Approximately Relativistic Lagrangians and the Center-of-Mass Theorem. I

In Newtonian mechanics the ten classical integrals of the equations of motion of a system of interacting point particles can be related to the invariance of the corresponding Lagrangian under the ten infinitesimal transformations of the Galilei group. Systems described by approximately relativistic equations [such as the Darwin equations in special relativity and the Einstein-Infeld-Hoffmann (EIH) equations in general relativity] also possess ten integrals of equivalent physical significance; previous work has established similar relations with invariance properties of the Lagrangian for only seven of these, but not for the three expressing the uniform motion of the center of mass. It is shown here that for any Lagrangian whatever which is a function of the particle positions and velocities alone, and which is invariant under the infinitestimal time and space translations, it is possible to find an additional exact invariance under a three-parameter set of infinitesimal transformations (which, in general, depends on a functional rather than a function). The transformations define a velocity which for approximately relativistic systems can be interpreted as that of their center of mass; for such systems the three conservation laws following from this transformation express the constancy of this velocity. A number of examples are given; for the Darwin and EIH equations, the conservation laws agree with those previously obtained directly from these equations.