Classical dynamics on triangleland

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.

Rewritten, publication reference added

References in corpus (16)

• Classical Equations for Quantum Systems
• Relativity without relativity
• The physical gravitational degrees of freedom
• Somewhere in the Universe: Where is the Information Stored When Histories Decohere?
• Scale-Invariant Gravity: Particle Dynamics
• Scale-Invariant Gravity: Geometrodynamics
• Trajectories for the Wave Function of the Universe from a Simple Detector Model
• Emergent Semiclassical Time in Quantum Gravity. I. Mechanical Models
• Interacting vector fields in Relativity without Relativity
• Strong-coupled Relativity without Relativity
• Variations on the Seventh Route to Relativity
• Relational Particle Models. II. Use as toy models for quantum geometrodynamics
• Relational Particle Models. I. Reconciliation with standard classical and quantum theory
• The geometry of the Barbour-Bertotti theories II. The three body problem
• The geometry of the Barbour-Bertotti theories I. The reduction process
• Relational particle models as toy models for quantum gravity and quantum cosmology

Cited by in corpus (16)

• Foundations of Relational Particle Dynamics
• 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
• Approaching the Problem of Time with a Combined Semiclassical-Records-Histories Scheme
• Triangleland. II. Quantum Mechanics of Pure Shape
• Jacobi's Principle and the Disappearance of Time
• On the Semiclassical Approach to Quantum Cosmology
• On the Conceptual Issues Surrounding the Notion of Relational Bohmian Dynamics
• Relational Motivation for Conformal Operator Ordering in Quantum Cosmology
• Quantum Cosmological Metroland Model
• Shape Space Methods for Quantum Cosmological Triangleland
• Quantum Cosmological Relational Model of Shape and Scale in 1-d
• A Novel Approach to Quantum Gravity in the Presence of Matter without the Problem of Time
• Towards the Unification of Gravity and other Interactions: What has been Missed?
• Relational Quadrilateralland. II. The Quantum Theory