Foundations of Relational Particle Dynamics

TL;DR

This paper explores the mathematical foundations of relational particle dynamics, focusing on shape and scale, and connects these to Barbour's relational theory, providing new insights into configuration spaces in low dimensions.

Contribution

It constructs natural mechanics on shape spaces like n-spheres and complex projective spaces, linking them to Barbour's relational dynamics and introducing a new theory with purely relative position and scale.

Findings

Configuration spaces identified as n-spheres and complex projective spaces.

Established equivalence between constructed mechanics and Barbour's relational dynamics.

Introduced a new theory with a more tractable configuration space for scale-invariant studies.

Abstract

Relational particle dynamics include the dynamics of pure shape and cases in which absolute scale or absolute rotation are additionally meaningful. These are interesting as regards the absolute versus relative motion debate as well as discussion of conceptual issues connected with the problem of time in quantum gravity. In spatial dimension 1 and 2 the relative configuration spaces of shapes are n-spheres and complex projective spaces, from which knowledge I construct natural mechanics on these spaces. I also show that these coincide with Barbour's indirectly-constructed relational dynamics by performing a full reduction on the latter. Then the identification of the configuration spaces as n-spheres and complex projective spaces, for which spaces much mathematics is available, significantly advances the understanding of Barbour's relational theory in spatial dimensions 1 and 2. I also provide the parallel study of a new theory for which positon and scale are purely relative but orientation is absolute. The configuration space for this is an n-sphere regardless of the spatial dimension, which renders this theory a more tractable arena for investigation of implications of scale invariance than Barbour's theory itself.