The Maximal Invariance Group of Newtons's Equations for a Free Point Particle

TL;DR

This paper identifies a larger symmetry group for Newton's equations of a free particle, extending the Galilei group with an $SL(2,R)$ component, and explores its implications for fluid mechanics and physical systems.

Contribution

It reveals the maximal invariance group of Newton's equations includes an $SL(2,R)$ component, connecting particle symmetries to fluid mechanics and physical phenomena.

Findings

The symmetry group extends the Galilei group with an $SL(2,R)$ component.

The $SL(2,R)$ symmetry explains similarities in supernova and plasma implosion simulations.

Examples of many-body systems exhibiting this symmetry are provided.

Abstract

The maximal invariance group of Newton's equations for a free nonrelativistic 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 {\it expansions}. This group was first discovered by Niederer in the context of the free Schr\"odinger 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 supernova explosions and numerical simulations of 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.