Small denominators and problems of stability of motion in classical and celestial mechanics. (English. Russian original) Zbl 0135.42701

Arnol’d, V. I.

Russ. Math. Surv. 18, No. 6, 85-191 (1963); translation from Usp. Mat. Nauk 18, No. 6(114), 91-192 (1963).

This paper gives complete proofs of the following results which have previously been published by the author [Sov. Math., Dokl. 3, 1008–1012 (1962); translation from Dokl. Akad. Nauk SSSR 145, 487–490 (1962; Zbl 0123.28101); preceding review Zbl 0135.42603]:
1) The stability of positions of equilibrium and periodic solutions of conservative systems with two degrees of freedom in the so-called general elliptic case.
2) The perpetual adiabatic invariance of the variable of action has been proved for a slow periodic variation of the parameters of a nonlinear oscillatory system with one degree of freedom. It has been established that a “magnetic trap” with an axial-symmetric magnetic field can perpetually retain charged particles.
3) Conditionally periodic motions in the many-body problem have been found. If the masses of \(n\) planets are sufficiently small in comparison with the mass of the central body, the motion is conditionally periodic for the majority of initial conditions for which the eccentricities and inclinations of the Kepler ellipses are small. Further, the major semiaxis perpetually remain close to their original values and the eccentricities and inclinations remain small.

Reviewer: F. Krückeberg

MSC:

70H14	Stability problems for problems in Hamiltonian and Lagrangian mechanics
70F15	Celestial mechanics
70H11	Adiabatic invariants for problems in Hamiltonian and Lagrangian mechanics
70H09	Perturbation theories for problems in Hamiltonian and Lagrangian mechanics
70H08	Nearly integrable Hamiltonian systems, KAM theory
70H07	Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics

Keywords:

small denominators; stability of motion; classical mechanics; celestial mechanics

Citations:

Zbl 0123.28101; Zbl 0135.42603