Partial Reduction in the NBody Planetary Problem using the Angular Momentum Integral
Abstract
We present a new set of variables for the reduction of the planetary nbody problem, associated to the angular momentum integral, which can be of any use for perturbation theory. The construction of these variables is performed in two steps. A first reduction, called partial is based only on the fixed direction of the angular momentum. The reduction can then be completed using the norm of the angular momentum. In fact, the partial reduction presents many advantages. In particular, we keep some symmetries in the equations of motion (d'Alembert relations). Moreover, in the reduced secular system, we can construct a Birkhoff normal form at any order. Finally, the topology of this problem remains the same as for the nonreduced system, contrarily to Jacobi's reduction where a singularity is present for zero inclinations. For three bodies, these reductions can be done in a very simple way in Poincaré's rectangular variables. In the general nbody case, the reduction can be performed up to a fixed degree in eccentricities and inclinations, using computer algebra expansions. As an example, we provide the truncated expressions for the change of variable in the 4body case, obtained using the computer algebra system TRIP.
 Publication:

Celestial Mechanics and Dynamical Astronomy
 Pub Date:
 November 2002
 Bibcode:
 2002CeMDA..84..283M
 Keywords:

 perturbation theory;
 reduction;
 Jacobi's reduction;
 resonance;
 lie transformation;
 computer algebra