On the co-existence of maximal and whiskered tori in the planetary three-body problem

Abstract

In this paper, we discuss about the possibility of the coexistence of stable and unstable quasi-periodic Kolmogorov–Arnold–Moser (kam) tori in a region of the phase space of the three-body problem. The argument of proof goes along kam theory and, especially, the production of two non-smoothly related systems of canonical coordinates in the same region of the phase space, the possibility of which is foreseen, for “properly degenerate” systems, by a theorem of Nekhoroshev and Miščenko and Fomenko. The two coordinate systems are alternative to the classical reduction of the nodes by Jacobi, described, e.g., in Arnold [“Small denominators and problems of stability of motion in classical and celestial mechanics,” 18, 85–191 (1963)].