August  2013, 33(8): 3555-3565. doi: 10.3934/dcds.2013.33.3555

On "Arnold's theorem" on the stability of the solar system

1. 

Université Paris-Dauphine, CEREMADE, Place du Maréchal de Lattre de Tassigny, Paris, France

Received  October 2012 Revised  November 2012 Published  January 2013

Arnold's theorem on the planetary problem states that, assuming that the masses of $n$ planets are small enough, there exists in the phase space a set of initial conditions of positive Lebesgue measure, leading to quasiperiodic motions with $3n-1$ frequencies. Arnold's initial proof is complete only for the plane $2$-planet problem. Arnold had missed a resonance later discovered by Herman. The first complete proof, by Herman-Féjoz, relies on the weak non-degeneracy condition of Arnold-Pyartli. A second proof, by Chierchia-Pinzari, is closer to Arnold's initial idea and shows the strong non-degeneracy of the problem after suitable reduction by (part of) the symmetry of rotation. We review and compare these proofs. In an appendix, we define the Poincaré coordinates and prove their symplectic nature through the shortest possible computation.
Citation: Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555
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Funkcional. Anal. i Priložen., 3 (1969), 59-62.  Google Scholar

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Celestial Mech. Dynam. Astronom., 62 (1995), 219-261. doi: 10.1007/BF00692089.  Google Scholar

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show all references

References:
[1]

Regul. Chaotic Dyn., 6 (2001), 421-432. doi: 10.1070/RD2001v006n04ABEH000186.  Google Scholar

[2]

Uspehi Mat. Nauk, 18 (1963), 91-192.  Google Scholar

[3]

Springer-Verlag, Berlin, 2006.  Google Scholar

[4]

Z. Angew. Math. Phys., 57 (2006), 33-41. doi: 10.1007/s00033-005-0002-0.  Google Scholar

[5]

Mem. Amer. Math. Soc., 187 (2007), viii+134 pp.  Google Scholar

[6]

Technical report, Bureau des Longitudes, 1989. Google Scholar

[7]

Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 545-578. doi: 10.3934/dcdss.2010.3.545.  Google Scholar

[8]

J. Mod. Dyn., 5 (2011), 623-664.  Google Scholar

[9]

Invent. Math., 186 (2011), 1-77. doi: 10.1007/s00222-011-0313-z.  Google Scholar

[10]

Ergodic Theory Dynam. Systems, 24 (2004), 1521-1582. doi: 10.1017/S0143385704000410.  Google Scholar

[11]

preprint, 2011, arXiv:1109.2892. Google Scholar

[12]

Duke Math. J., 159 (2011), 275-327. doi: 10.1215/00127094-1415878.  Google Scholar

[13]

Bulletin Astronomique, 3 (1966), 49-66. Google Scholar

[14]

Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.  Google Scholar

[15]

Icarus, 88 (1990), 266-291. Google Scholar

[16]

in "Le Chaos," Séminaire Poincaré, XIV, Birkhäuser, (2010), 221-246. Google Scholar

[17]

Celestial Mech. Dynam. Astronom., 62 (1995), 193-217. doi: 10.1007/BF00692088.  Google Scholar

[18]

Celestial Mech., 13 (1976), 471-489.  Google Scholar

[19]

Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 377-398 (electronic). doi: 10.3934/dcdsb.2007.7.377.  Google Scholar

[20]

Celestial Mechanics and Dynamical Astronomy, 84 (2002), 283-316. doi: 10.1023/A:1020392219443.  Google Scholar

[21]

in "Hamiltonian Dynamical Systems" (Boulder, CO, 1987), Contemp. Math., 81, Amer. Math. Soc., Providence, RI, (1988), 1-22. doi: 10.1090/conm/081/986254.  Google Scholar

[22]

Icarus, 8 (1968), 203-215. Google Scholar

[23]

With special emphasis on celestial mechanics, Reprint of the 1973 original, With a foreword by Philip J. Holmes, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.  Google Scholar

[24]

New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[25]

Trudy Sem. Petrovsk., 5 (1979), 5-50.  Google Scholar

[26]

Dedicated to V. I. Arnold on the occasion of his 65th birthday, Mosc. Math. J., 3 (2003), 1039-1052, 1200.  Google Scholar

[27]

in "Hamiltonian Dynamical Systems and Applications," NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, (2008), 53-66. doi: 10.1007/978-1-4020-6964-2_3.  Google Scholar

[28]

Nonlinearity, 9 (1996), 1703-1751. doi: 10.1088/0951-7715/9/6/017.  Google Scholar

[29]

Ph.D thesis, Universitá di Roma Tre, 2009. Google Scholar

[30]

Librairie Scientifique et Technique Albert Blanchard, Gauthier-Villars, Paris, 1892.  Google Scholar

[31]

Gauthier-Villars, 1905. Google Scholar

[32]

Funkcional. Anal. i Priložen., 3 (1969), 59-62.  Google Scholar

[33]

Celestial Mech. Dynam. Astronom., 62 (1995), 219-261. doi: 10.1007/BF00692089.  Google Scholar

[34]

Phys. D, 140 (2000), 1-32. doi: 10.1016/S0167-2789(99)00211-0.  Google Scholar

[35]

Die Grundlehren der mathematischen Wissenschaften, Band 174, Springer-Verlag, New York-Heidelberg, 1971.  Google Scholar

[36]

Gauthier-Villars, Paris, 1896. Google Scholar

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