Invariant relationships deriving from classical scaling transformations

TL;DR

This paper explores how classical scaling transformations, though not variational symmetries, lead to evolutionary laws that connect physical quantities in dynamical and static systems, revealing generalized virial and energy relations.

Contribution

It extends Noether's theorem to include scaling transformations, deriving evolutionary laws that relate kinematic and dynamic variables in various physical systems.

Findings

Derives a scaling evolutionary law applicable to dynamical and static systems.

Reveals generalized virial laws linking kinetic and potential energies.

Explains properties of polytropes in stellar models.

Abstract

Because scaling symmetries of the Euler-Lagrange equations are generally not variational symmetries of the action, they do not lead to conservation laws. Instead, an extension of Noether's theorem reduces the equations of motion to evolutionary laws that prove useful, even if the transformations are not generalized symmetries of the equations of motion. In the case of scaling, symmetry leads to a scaling evolutionary law, a first-order equation in terms of scale invariants, linearly relating kinematic and dynamic degrees of freedom. This scaling evolutionary law appears in dynamical and in static systems. Applied to dynamical central-force systems, the scaling evolutionary equation leads to generalized virial laws, which linearly connect the kinetic and potential energies. Applied to barotropic hydrostatic spheres, the scaling evolutionary equation linearly connects the gravitational and internal energy densities. This implies well-known properties of polytropes, describing degenerate stars and chemically homogeneous non-degenerate stellar cores.