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

*2020, Volume 25, Number 2, pp. 149-165*

Author(s):

**Lim C. C.**

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
$T_c(5)$.

Access to the full text on the Springer website |