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 $N = 2 d + 1$, 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.