Euclidean and similarity relational particle mechanics: classical theory for 3 particles in 2D

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. 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 itself standard universal large-relative-scale behaviour. What is unusual about SRPM is the transition between these two behaviours: it is a model in which a symmetry principle underlies an unexpected departure from standard physical behaviour at sufficiently large relative scales. That has the scope, at least qualitatively, to account for large-relative-scale deviations from standard physics as might e.g. account for galaxy rotation curves. 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