The Connection between Conservation Laws and Laws of Motion in Affine Spaces

It was pointed out recently that, for any theory describing matter as a collection of mass points in a metric space and subject to a covariant conservation law for a symmetric tensor density $\mathfrak{P}{}^{\mathrm{\mu \; \nu }}$ , the geodesic law of motion as well as the form of $\mathfrak{P}{}^{\mathrm{\mu \; \nu }}$ follow from the conservation law alone, independent of any equations obeyed by the metric. This result is shown to be valid in any affine space, independent of any equations obeyed by the affine connection; conversely, the geodesic law implies a conservation law for a singular symmetric tensor density. Similarly, the existence in any affine space of a covariant conservation law for a vector density $\mathfrak{F}{}^{\nu }$ describing a collection of point charges is shown to imply the constancy of charge, and the form of $\mathfrak{F}{}^{\nu }$ ; conversely, the constancy of charge implies a conservation law for a singular vector density. Some applications of these results are presented. An Appendix contains a discussion of the laws of motion for particles with an intrinsic dipole moment.