The geometry of the Barbour-Bertotti theories I. The reduction process

Abstract

The dynamics of $N\ge 3$ interacting particles is investigated in the non-relativistic context of the Barbour-Bertotti theories. The reduction process on this constrained system yields a Lagrangian in the form of a Riemannian line element. The involved metric, degenerate in the flat configuration space, is the first fundamental form of the space of orbits of translations and rotations (the Leibniz group). The Riemann tensor and the scalar curvature are computed by a generalized Gauss formula in terms of the vorticity tensors of generators of the rotations. The curvature scalar is further given in terms of the principal moments of inertia of the system. Line configurations are singular for $N\ne 3$. A comparison with similar methods in molecular dynamics is traced.

15 pages, to appear in Classical and Quantum Gravity

References in corpus (3)

• The Generalized Thin-Sandwich Problem and its Local Solvability
• The geometry of the Barbour-Bertotti theories II. The three body problem
• Classical Mechanics without Absolute Space

Cited by in corpus (21)

• Foundations of Relational Particle Dynamics
• Emergent Semiclassical Time in Quantum Gravity. I. Mechanical Models
• 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
• Relational Particle Models. I. Reconciliation with standard classical and quantum theory
• Relational Particle Models. II. Use as toy models for quantum geometrodynamics
• Records Theory
• The geometry of the Barbour-Bertotti theories II. The three body problem
• Triangleland. II. Quantum Mechanics of Pure Shape
• Classical dynamics on triangleland
• Jacobi's Principle and the Disappearance of Time
• On the Semiclassical Approach to Quantum Cosmology
• Shape dynamics and Mach's principles: Gravity from conformal geometrodynamics
• Quantum Machian Time in Toy Models of Gravity
• Implementing Mach's Principle Using Gauge Theory
• Relational Motivation for Conformal Operator Ordering in Quantum Cosmology
• On the Conceptual Issues Surrounding the Notion of Relational Bohmian Dynamics
• Quantum Cosmological Metroland Model
• Shape Space Methods for Quantum Cosmological Triangleland
• Quantum Cosmological Relational Model of Shape and Scale in 1-d
• Relational Quadrilateralland. II. The Quantum Theory