$N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities

Abstract

We review the $N$-Body Problem in arbitrary dimension $d$ at the kinematical level, with modelling Background Independence in mind. In particular, we give a structural analysis of its reduced configuration spaces, decomposing this subject matter into basic Topology, Geometry, Group Theory, Linear Algebra, Graph Theory and Order Theory. At the metric level, these configuration spaces of shapes form basic geometric series for 1- and 2-$d$: ${\mathbb{S}}^{N-2}$ and ${\mathbb{CP}}^{N-2}$, though there are no more such series for $d\ge 3$. $d\ge 3$ also sees an onset of stratification. Casson's diagonal, for which $N=d+1$, plays a critical role which we explain in simple Linear Algebra terms; these are moreover topologically spheres. $N=d$ and $N=d+2$ have further significance as well, the latter as regards a counting notion of genericity of the isotropy groups and kinematical orbits realized. These observations twin the $N$ = (3, 4, 5) progression in qualitative complexity that almost all $N$-Body Problem work for concrete $N$ concentrates on with the much better-known $N$ = (2, 3, 4) progression in 2-$d$: from intervals to triangles to quadrilaterals: a large source of intuitions and mathematical analogies. We furthermore provide an Order-Theoretic genericity criterion, which is almost always bounded by the double-slope $N=2d+1$ line, though an accidental relation pushes $N$ up by 1 to 8 in 3-$d$. We finally consider rubber shapes, for which the configuration spaces are graphs: much simpler than stratified manifolds, and yet containing quite a few of metric-level shapes' qualitative features. This singles out $N=5$ and 6 in 1-$d$ and $N=6$ and 8 in $\ge$ 2-$d$ for the onsets of various graph-theoretical nontrivialities.

59 pages including 27 figures

References in corpus (21)

• Some remarks on the notions of general covariance and background independence
• The Superspace of Geometrodynamics
• Beables/Observables in Classical and Quantum Gravity
• Foundations of Relational Particle Dynamics
• Problem of Time and Background Independence: the Individual Facets
• Explicit partial and functional differential equations for beables or observables
• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• The Smallest Shape Spaces. III. Triangles in 2- and 3-d
• The Smallest Shape Spaces. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity
• Relational mechanics of shape and scale
• Configuration Spaces in Fundamental Physics
• Manifolds of Projective Shapes
• Alice in Triangleland: Lewis Carroll's Pillow Problem and Variants Solved on Shape Space of Triangles
• Two new versions of Heron's Formula
• Monopoles of Twelve Types in 3-Body Problems
• Maximal Angle Flow on the Shape Sphere of Triangles
• Shape (In)dependent Inequalities for Triangleland's Jacobi and Democratic-Linear Ellipticity Quantitities
• Absolute versus Relational Debate: a Modern Global Version
• Topological Shape Theory
• Background Independence: ${\mathbb{S}}^1$ and $\mathbb{R}$ absolute spaces differ greatly in Shape-and-Scale Theory
• Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces

Cited by in corpus (10)

• Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups
• A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism
• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• Geometry from Brackets Consistency
• Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups
• A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics
• Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups
• Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations
• A Local Resolution of the Problem of Time. VIII. Assignment of Observables
• Isotropy Groups and Kinematic Orbits for 1 and 2-$d$ $N$-Body Problems