Mass Points

Abstract

A dynamics is a representation of spacetime operations. A nonrelativistic mass point dynamics is a representation of the time translations. The equations of motion, of second order in time for position as basic observable and of first order for the position-momentum pair, express the Lie algebra action $\frac{d}{{dt}}$ of the time operations. The classical mass point orbits as solutions are realizations of the (eigen)time translation group $t \in \mathcal{R} \cong D\left( 1 \right) \ni e^t$ in position, faithful noncompact with image $\mathcal{R} \cong {\bf SO}_0 \left( {1,1} \right) = y^1$,e.g., for free mass points and hyperbolic orbits of never-returning comets, and unfaithful compact with image $\mathcal{R}/ \mathcal{Z} \cong {\bf SO}\left( 2 \right) = \Omega ^1$ for periodic orbits, e.g., for elliptic orbits of planets. Newton’s idealization of mechanics working with mass points was successful even after the introduction of the electromagnetic fields by Faraday and Maxwell, and of the metrical tensor field in Einstein’s gravity. The time orbits of mass points in position are derivable by an extremalization of an action, leading to the Euler–Lagrange equations of motion. For a general relativistic mass point dynamics, this extremalization merges into the property of geodesics to have an extremal length; the Lagrangian is essentially the spacetime metric.