Multiple-time constants of the motion for manifestly Poincaré-invariant Newtonian-like equations of motion for point particles Available to Purchase

While it is known that manifest Poincaré invariance of equations of motion for an isolated system of $N$ interacting point particles is not a guarantee of conserved quantities, here it is shown that if such equations of motion have a Newtonian-like form (e.g., the formalism of Havas and Plebański), then ten multiple-time constants of the motion can be constructed by direct integration of the equations of motion. That these directly integrated quantities may be regarded as the ten standard relativistic multiple-time conserved quantities is verified by showing that a known subset, the Lagrangian-based multiple-time conserved quantities that are usually derived from symmetries of manifestly Poincaré-invariant variational principles, here can be obtained by substitution of the Lagrangian forces (which are read from the Newtonian-like Lagrangian equations of motion) into the directly integrated multiple-time forms. This method of obtaining relativistic multiple-time conserved quantities shows that, contrary to some previous statements in the literature, a manifestly Poincaré-invariant “Newton’s third law of motion” is not necessary to guarantee the existence of a conserved four-momentum. Furthermore, an analogous calculation in Newtonian mechanics (using a Galilei-invariant integration parameter instead of a Poincaré-invariant one) yields Newtonian multiple-time constants of the motion. The standard one-time forms of Newtonian conserved quantities do follow from these Newtonian multiple-time constants of the motion when all $N$ times are chosen to be equal and the two-body forces follow from a sum of particle-symmetric two-body potential energies depending only on the mutual interparticle separations. A main result of this direct integration method of obtaining classical relativistic conserved quantities (which involve integrals over the whole motion) is that such quantities are far more analogous to the Newtonian multiple-time constants of the motion (which are just integrals of the equations of motion, independent of any symmetries of these equations) than to the standard Newtonian conservation laws (which apply only under restricted conditions).