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 U 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).