zbMATH — the first resource for mathematics

Analytic Lagrangian tori for the planetary many-body problem. (English) Zbl 1173.37067
The planetary many-body problem consists of studying the evolution of \((n+1)\) bodies (point masses), subject only to mutual gravitation attraction, in the case where one of the bodies (the ‘Sun’) has a mass considerably larger than the masses of the remaining \(n\) bodies (the ‘planets’). The invariant tori associated with the motions provided by the well-known Arnold Theorem, in view of KAM tools, are smooth. Now, since the many-body problem is formulated in terms of real-analytic functions, it appears somewhat more natural to seek for real-analytic invariant manifolds. This is the problem addressed in this paper and, in particular, a new proof of Arnold’s statement is obtained.

37N05 Dynamical systems in classical and celestial mechanics
70F10 \(n\)-body problems
70F15 Celestial mechanics
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
Full Text: DOI arXiv
[1] DOI: 10.1017/S0143385704000410 · Zbl 1087.37506 · doi:10.1017/S0143385704000410
[2] DOI: 10.1017/S0143385704000410 · Zbl 1087.37506 · doi:10.1017/S0143385704000410
[3] Salamon, Math. Phys. Electron. J. 3 (2004)
[4] Celletti, Mem. Amer. Math. Soc. 187 pp 134– (2007)
[5] Sevryuk, Mosc. Math. J. 3 pp 1113– (2003)
[6] Rüßmann, R. & C. Dynamics 2 pp 119– (2001)
[7] DOI: 10.1017/S0143385706000885 · Zbl 1130.37376 · doi:10.1017/S0143385706000885
[8] DOI: 10.1007/BF00692089 · Zbl 0837.70009 · doi:10.1007/BF00692089
[9] DOI: 10.1007/BF01228562 · Zbl 0517.70015 · doi:10.1007/BF01228562
[10] DOI: 10.1007/BF01076316 · Zbl 0216.04401 · doi:10.1007/BF01076316
[11] DOI: 10.1007/s00205-003-0269-2 · Zbl 1036.70006 · doi:10.1007/s00205-003-0269-2
[12] Mathematical Aspects of Classical and Celestial Mechanics (2006)
[13] Poincaré, Leçons de mécanique céleste (1905)
[14] Arnol’d, Uspekhi Mat. Nauk. 18 pp 91– (1963)
[15] Laskar, Predictability, Stability and Chaos in pp 93– (1991) · doi:10.1007/978-1-4684-5997-5_7
[16] Hofer, Symplectic Invariants and Hamiltonian Dynamics (1994) · doi:10.1007/978-3-0348-8540-9
[17] DOI: 10.1007/BF01234305 · Zbl 0623.70010 · doi:10.1007/BF01234305
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.