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Mitchell and Littlejohn showed that isotropy groups and orbits for $N$-bodyproblems attain a sense of genericity for $N = 5$. The author recently showedthat the arbitrary-$d$ generalization of this 3-$d$ result is that genericityin this sense occurs for $N = d + 2$. The author also showed that a secondsense 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 inpassing from 3 to 4 and 5 body problems in 3-$d$ is already present in themore-well known setting of passing from intervals to triangles and then toquadrilaterals in 2-$d$. 2) That not $(d, N) = (3, 6)$ but $(4, 6)$ is anatural theoretical successor of $(3, 5)$. 3) Such consideration isotropygroups and orbits is moreover a model for a larger case of interest, namelythat of GR's reduced configuration spaces. The current Article presents thelower-$d$ cases explicitly: 0, 1 and 2-$d$, including also the topological andgeometrical form of the corresponding isotropy groups and orbits.

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