KAM stability and celestial mechanics.

*(English)*Zbl 1129.70012
Mem. Am. Math. Soc. 878, 134 p. (2007).

Summary: KAM theory is a powerful tool apt to prove perpetual stability in Hamiltonian systems, which are a perturbation of integrable ones. The smallness requirements for its applicability are well known to be extremely stringent. A long standing problem, in this context, is the application of KAM theory to “physical systems” for “observable” values of perturbation parameters. We consider the restricted circular planar three-body problem (RCP3BP), i.e., the problem of studying the planar motions of a small body subject to gravitational attraction of two primary bodies revolving on circular Keplerian orbits (which are assumed not to be influenced by the small body). When the mass ratio of the two primary bodies is small, the RCP3BP is described by a nearly-integrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with non-small eccentricities, the system is well described by Delaunay variables. The Sun-Jupiter observed motion is nearly circular, and an asteroid of the asteroidal belt may be assumed not to influence the Sun-Jupiter motion. The Jupiter-Sun mass ratio is slightly less than 1/1000. We consider the motion of the asteroid 12 Victoria taking into account only the Sun-Jupiter gravitational attraction regarding such a system as a prototype of RCP3BP. For values of mass ratios up to 1/1000, we prove the existence of two-dimensional KAM tori on a fixed three-dimensional energy level corresponding to the observed energy of Sun-Jupiter-Victoria system. Such tori trap the evolution of phase points “close” to the observed physical data of Sun-Jupiter-Victoria system. As a consequence, in the RCP3BP description, the motion of Victoria is proven to be forever close to an elliptical motion. The proof is based on: 1) a new iso-energetic KAM theory; 2) an algorithm for computing iso-energetic, approximate Lindstedt series; 3) a computer-aided application of 1) and 2) to the Sun-Jupiter-Victoria system. The paper is self-contained but does not include the (\(\sim\) 12000 lines) computer programs, which may be obtained by sending an e-mail to one of the authors.

##### MSC:

70F15 | Celestial mechanics |

70F07 | Three-body problems |

70H08 | Nearly integrable Hamiltonian systems, KAM theory |

70H09 | Perturbation theories for problems in Hamiltonian and Lagrangian mechanics |

37N05 | Dynamical systems in classical and celestial mechanics |