Newton ’ S Equations for a Free Point Particle †

The maximal invariance group of Newton's equations for a free non-relativistic point particle is shown to be larger than the Galilei group. It is a semi-direct product of the static (nine-parameter) Galilei group and an SL(2, R) group containing time-translations, dilations and a one-parameter group of time-dependent scalings called expansions. This group was first discovered by Niederer in the context of the free Schrödinger equation. We also provide a road map from the free nonrelativistic point particle to the equations of fluid mechanics to which the symmetry carries over. The hitherto unnoticed SL(2, R) part of the symmetry group for fluid mechanics gives a theoretical explanation for an observed similarity between numerical simulations of su-pernova explosions and experiments involving laser-induced implosions in inertial confinement plasmas. We also give examples of interacting many body systems of point particles which have this symmetry group.