The planetary N-body problem: Symplectic foliation, reductions and invariant tori

The 6n-dimensional phase space of the planetary (1 + n)-body problem (after the classical reduction of the total linear momentum) is shown to be foliated by symplectic leaves of dimension (6n − 2) invariant for the planetary Hamiltonian H. Such foliation is described by means of a new global set of Darboux coordinates related to a symplectic (partial) reduction of rotations. On each symplectic leaf H has the same form and it is shown to preserve classical symmetries. Further sets of Darboux coordinates may be introduced on the symplectic leaves so as to achieve a complete (total) reduction of rotations. Next, by explicit computations, it is shown that, in the reduced settings, certain degeneracies are removed. In particular, full torsion is checked both in the partially and totally reduced settings. As a consequence, a new direct proof of Arnold’s theorem (Arnold in Russ. Math. Surv. 18(6):85– 191, 1963) on the stability of planetary system (both in the partially and in the totally reduced setting) is easily deduced, producing Diophantine Lagrangian invariant tori of dimension (3n − 1) and (3n − 2). Finally, elliptic lower dimensional tori bifurcating from the secular equilibrium are easily obtained.