General Jacobi Coordinates and Herman Resonance for Some Nonheliocentric Celestial $N$-body Problems

The general Jacobi symplectic variables generated by a combinatorial algorithm from the full binary tree $T(N)$ are used to formulate some nonheliocentric gravitational $N$-body problems in perturbation form. The resulting uncoupled term $H_U$ for $(N-1)$ independent Keplerian motions and the perturbation term $H_P$ are both explicitly dependent on the partial ordering induced by the tree $T(N)$. This leads to suitable conditions on separations of the $N$ bodies for the perturbation to be small. We prove the Herman resonance for a new approximation of the 5-body problem. Full details of the derivations of the perturbation form and Herman resonance are given only in the case of five bodies using the caterpillar binary tree $T_c(5)$.