Isotropy Groups and Kinematic Orbits for 1 and 2-$d$ $N$-Body Problems

TL;DR

This paper explores the structure and genericity of isotropy groups and kinematic orbits in N-body problems across different dimensions, revealing patterns and applications relevant to both classical mechanics and general relativity.

Contribution

It generalizes previous results on isotropy groups in N-body problems to arbitrary dimensions and analyzes their topological and geometric properties in low dimensions.

Findings

Genericity occurs at N=d+2 for arbitrary-d N-body problems.

In 2D, isotropy groups and orbits are explicitly characterized.

Applications include insights into complexity transitions and GR configuration spaces.

Abstract

Mitchell and Littlejohn showed that isotropy groups and orbits for $N$-body problems attain a sense of genericity for $N = 5$. The author recently showed that the arbitrary-$d$ generalization of this 3-$d$ result is that genericity in this sense occurs for $N = d + 2$. The author also showed that a second sense of genericity -- now order-theoretic rather than a matter of counting -- occurs for $N = 2 d + 1$, excepting $d = 3$, for which it is not 7 but 8. Applications of this work include 1) that some of the increase in complexity in passing from 3 to 4 and 5 body problems in 3-$d$ is already present in the more-well known setting of passing from intervals to triangles and then to quadrilaterals in 2-$d$. 2) That not $(d, N) = (3, 6)$ but $(4, 6)$ is a natural theoretical successor of $(3, 5)$. 3) Such consideration isotropy groups and orbits is moreover a model for a larger case of interest, namely that of GR's reduced configuration spaces. The current Article presents the lower-$d$ cases explicitly: 0, 1 and 2-$d$, including also the topological and geometrical form of the corresponding isotropy groups and orbits.