Classical dynamics on triangleland

TL;DR

This paper explores classical dynamics of three particles in 2D under Euclidean and similarity relational mechanics, providing exact solutions and analyzing models with Newton-Coulomb and harmonic potentials, highlighting unique large-scale behaviors.

Contribution

It offers a fully reduced classical framework for triangleland in 2D for ERPM and SRPM, including exact solutions and numerical analysis of specific models, revealing novel large-scale behavior in SRPM.

Findings

Exact solutions for ERPM and SRPM in 2D triangleland.

SRPM exhibits universal large-relative-scale behavior.

Models show standard small-scale and unexpected large-scale dynamics.

Abstract

In Euclidean relational particle mechanics (ERPM) only relative times, relative angles and relative separations are meaningful, while in similarity relational particle mechanics (SRPM) only relative times, relative angles and ratios of relative separations are. These theories are clearly of interest in the absolute or relative motion debate. In this paper, ERPM and SRPM are provided in fully reduced form for 3 particles in 2D, i.e. the classical dynamics on triangleland in 2D with and without scale. Exact solutions to each of these are then found, and simple Newton--Coulomb-like and harmonic oscillator-like SRPM models are studied numerically. The mathematics one arrives at thus overlaps in many ways with that which arises in the absolutist approach. The ERPM gives standard mathematics, while the SRPM has standard small-relative-scale behaviour and an unexpected but in itself standard universal large-relative-scale behaviour. One way in which SRPM is unusual is that it is a model in which a symmetry principle underlies an unexpected departure from standard physical behaviour at sufficiently large relative scales (interpolation between the abovementioned two behaviours). ERPM and SRPM are also theoretically interesting at the quantum level, both on their own merit and as toy models for the development of various approaches to the problems of time and of observables in quantum general relativity.