Foundations of Relational Particle Dynamics

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.

Minor typos corrected; references updated

References in corpus (7)

• New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split
• Triangleland. I. Classical dynamics with exchange of relative angular momentum
• Does relationalism alone control geometrodynamics with sources?
• Geometrodynamics: Spacetime or Space ?
• Classical dynamics on triangleland
• Seminar on Records Theory
• Geometrodynamics in a spherically symmetric, static crossflow of null dust

Cited by in corpus (5)

• New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split
• Triangleland. I. Classical dynamics with exchange of relative angular momentum
• Records Theory
• Does relationalism alone control geometrodynamics with sources?
• Triangleland. II. Quantum Mechanics of Pure Shape