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Analytic Lagrangian tori for the planetary many-body problem

Published online by Cambridge University Press:  01 June 2009

LUIGI CHIERCHIA
Affiliation:
Dipartimento di Matematica, Università ‘Roma Tre’, Largo S. L. Murialdo 1, I-00146 Roma, Italy (email: luigi@mat.uniroma3.it)
FABIO PUSATERI
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA (email: pusateri@cims.nyu.edu)

Abstract

In 2004, Féjoz [Démonstration du ‘théoréme d’Arnold’ sur la stabilité du système planétaire (d’après M. Herman). Ergod. Th. & Dynam. Sys.24(5) (2004), 1521–1582], completing investigations of Herman’s [Démonstration d’un théoréme de V.I. Arnold. Séminaire de Systémes Dynamiques et manuscripts, 1998], gave a complete proof of ‘Arnold’s Theorem’ [V. I. Arnol’d. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6(114)) (1963), 91–192] on the planetary many-body problem, establishing, in particular, the existence of a positive measure set of smooth (C) Lagrangian invariant tori for the planetary many-body problem. Here, using Rüßmann’s 2001 KAM theory [H. Rüßmann. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. R. & C. Dynamics2(6) (2001), 119–203], we prove the above result in the real-analytic class.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Arnol’d, V. I.. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6 (114)) (1963), 91192.Google Scholar
[2] V. I. Arnol’d, V. V. Kozlov and A. I. Neishtadt (eds). Mathematical Aspects of Classical and Celestial Mechanics, 3rd edition(Encyclopaedia of Mathematical Sciences, 3). Springer, Berlin, 2006.Google Scholar
[3]Biasco, L., Chierchia, L. and Valdinoci, E.. Elliptic two-dimensional invariant tori for the planetary three-body problem. Arch. Ration. Mech. Anal. 170 (2003), 91135.CrossRefGoogle Scholar
[4]Boigey, F.. Élimination des nòe uds dans le problème newtonien des quatre corps. Celestial Mech. 27(4) (1982), 399414.Google Scholar
[5]Broer, H. and Takens, F.. Unicity of Kam tori. Ergod. Th. & Dynam. Sys. 27 (2007), 713724.Google Scholar
[6]Celletti, A. and Chierchia, L.. Kam stability and celestial mechanics. Mem. Amer. Math. Soc. 187(878) (2007), 134.Google Scholar
[7]Deprit, A.. Elimination of the nodes in problems of n bodies. Celestial Mech. 30(2) (1983), 181195.Google Scholar
[8]Fathi, A., Giuliani, A. and Sorrentino, A.. Uniqueness of invariant Lagrangian graphs in a homology or a cohomology class, 2008. Preprint. Available at http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.3568v1.pdf.Google Scholar
[9]Féjoz, J.. Démonstration du ‘théoréme d’Arnold’ sur la stabilité du système planétaire (d’après M. Herman). Ergod. Th. & Dynam. Sys. 24(5) (2004), 15211582.CrossRefGoogle Scholar
[10]Féjoz, J.. Version révisée de l’article paru dans le Michael Herman Memorial Issue. Ergod. Th. & Dynam. Sys. 24(5) (2004), 15211582.Google Scholar
[11]Herman, M. R.. Démonstration d’un théoréme de V.I. Arnold. Séminaire de Systémes Dynamiques et manuscripts, Université D. Diderot, Paris 7, 1998.Google Scholar
[12]Hofer, H. and Zehnder, E.. Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, Basel, 1994.CrossRefGoogle Scholar
[13]Laskar, J.. Analytical framework in Poincaré variables for the motion of the solar system. Predictability, Stability and Chaos in n-Body Dynamical Systems (Nato Adv. Sci. Inst. Ser. B Phys., 272). Plenum, New York, 1991, pp. 93114.CrossRefGoogle Scholar
[14]Poincaré, H.. Leçons de mécanique céleste. Gauthier-Villars, Paris, 1905 (Vol. 1); 1907 (Vol. 2); 1910 (Vol. 3); 2005 (Reprint).Google Scholar
[15]Pusateri, F.. Analytic KAM tori for the planetary (n+1)-body problem. Master’s Thesis, Università degli Studi Roma Tre, 2006. http://www.mat.uniroma3.it/users/chierchia/TESI.Google Scholar
[16]Pyartli, A. S.. Diophantine approximations on Euclidean submanifolds. Functional. Anal. Appl. 3 (1969), 303306.Google Scholar
[17]Robutel, P.. Stability of the planetary three-body problem II. KAM theory and existence of quasi-periodic motions. Celestial Mech. Dynam. Astronom. 62(3) (1995), 219261.CrossRefGoogle Scholar
[18]Rüßmann, H.. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. R. & C. Dynamics 2(6) (2001), 119203.Google Scholar
[19]Salamon, D.. The Kolmogorov–Arnold–Moser theorem. Math. Phys. Electron. J. 3(37) (2004), (electronic).Google Scholar
[20]Sevryuk, M. B.. The classical KAM theory and the dawn of the twenty-first century. Mosc. Math. J. 3(3) (2003), 11131144.Google Scholar