Scale invariance and constants of motion

TL;DR

This paper explores how scale invariance in self-similar spacetimes relates to constants of motion for particles, deriving conditions and explicit forms of these conserved quantities for both massless and massive particles.

Contribution

It establishes a framework for understanding scale invariance and associated conserved quantities in self-similar spacetimes, including explicit solutions for these constants.

Findings

Constants of motion are characterized for massless and massive particles.

A conservation law holds on the phase space constraint surface.

Explicit forms of the constants of motion are derived.

Abstract

Scale invariance in the theory of classical mechanics can be induced from the scale invariance of background fields. In this paper we consider the relation between the scale invariance and the constants of particle motion in a self-similar spacetime, only in which the symmetry is well-defined and is generated by a homothetic vector. Relaxing the usual conservation condition by the Hamiltonian constraint in a particle system, we obtain a conservation law holding only on the constraint surface in the phase space. By the conservation law, we characterize constants of motion associated with the scale invariance not only for massless particles but for massive particles and classify the condition for the existence of the constants of motion. Furthermore, we find the explicit form of the constants of motion by solving the conservation equations.