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

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 $N=2d+1$, 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.

arXiv comment16 pages, including 4 figures. arXiv admin note: text overlap with arXiv:1807.08391

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