Reduction of Lagrangian Equations of Motion of Modified Newtonian Theory of Gravity with respect to the Similarity Group

TL;DR

This paper derives equations of motion for the shape of a classical system under modified Newtonian gravity, eliminating dependence on absolute position, orientation, or size, using symmetry reduction techniques.

Contribution

It introduces a method to reduce Lagrangian equations of motion to shape space for systems under similarity transformations, enabling shape evolution predictions without referencing absolute configurations.

Findings

Derived equations of motion on shape space for N-body systems.

Illustrated shape equations for a three-body system.

Established a metric on shape space from the configuration space metric.

Abstract

The equivalence class of absolute configurations of a system under the group of similarity transformations $Sim(3)$ is called the shape of the system. The $Sim(3)$ invariant Lagrangian of the modified Newtonian theory ensures the existence of the its law of motion on shape space. To deduce the equations of motion for a system's shape degrees of freedom from its evolution equations for the $3N$ absolute configuration degrees of freedom, the Boltzman-Hamel equations of motion in an non-holonomic frame on the tangent space $T(Q)$ to the system's absolute configuration space $Q$ is adapted to the $Sim(3)$ fiber bundle structure of the configuration space. The derived equations of motion on shape space enable us, among other things, to predict the evolution of the shape of a classical system governed by this theory without any reference to its absolute position, orientation, or size in space. The paper will explain, that by treating the measuring instruments as part of the matter in the theory, how the mass metric $\textbf{M}$ on the configuration space $Q$ uniquely defines a metric on the reduced tangent bundle $\frac{T(Q)}{Sim(3)}$, and how the unique metric structure on shape space $S$ can be derived. After deriving the reduced equations of motion on shape space for a general $N$-body system, the shape equations of motion for a three-body system in suitable coordinates is given as an illustration.