# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555
##### References:
 [1] K. Abdullah and A. Albouy, On a strange resonance noticed by M. Herman, Regul. Chaotic Dyn., 6 (2001), 421-432. doi: 10.1070/RD2001v006n04ABEH000186. [2] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, 18 (1963), 91-192. [3] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects Of Classical and Celestial Mechanics," Translated from the Russian original by E. Khukhro, Third edition, Encyclopaedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 2006. [4] A. Celletti and L. Chierchia, KAM stability for a three-body problem of the solar system, Z. Angew. Math. Phys., 57 (2006), 33-41. doi: 10.1007/s00033-005-0002-0. [5] A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc., 187 (2007), viii+134 pp. [6] A. Chenciner, Intégration du problème de Kepler par la méthode de Hamilton-Jacobi, Technical report, Bureau des Longitudes, 1989. [7] L. Chierchia and G. Pinzari, Properly-degenerate KAM theory (following V. I. Arnold), Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 545-578. doi: 10.3934/dcdss.2010.3.545. [8] L. Chierchia and G. Pinzari, Planetary Birkhoff normal forms, J. Mod. Dyn., 5 (2011), 623-664. [9] L. Chierchia and G. Pinzari, The planetary $N$-body problem: Symplectic foliation, reductions and invariant tori, Invent. Math., 186 (2011), 1-77. doi: 10.1007/s00222-011-0313-z. [10] J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), Ergodic Theory Dynam. Systems, 24 (2004), 1521-1582. doi: 10.1017/S0143385704000410. [11] J. Féjoz, M. Guàrdia, V. Kaloshin and P. Roldán, Diffusion along mean motion resonance in the restricted planar three-body problem, preprint, 2011, arXiv:1109.2892. [12] J. Galante and V. Kaloshin, Destruction of invariant curves in the restricted circular planar three-body problem by using comparison of action, Duke Math. J., 159 (2011), 275-327. doi: 10.1215/00127094-1415878. [13] M. Hénon, Exploration numérique du problème restreint IV. masses égales, orbites non périodiques, Bulletin Astronomique, 3 (1966), 49-66. [14] A. N. Kolmogorov, On the conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530. [15] J. Laskar, The chaotic motion of the solar system. a numerical estimate of the size of the chaotic zones, Icarus, 88 (1990), 266-291. [16] J. Laskar, Le système solaire est-il stable?, in "Le Chaos," Séminaire Poincaré, XIV, Birkhäuser, (2010), 221-246. [17] J. Laskar and P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian, Celestial Mech. Dynam. Astronom., 62 (1995), 193-217. doi: 10.1007/BF00692088. [18] M. L. Lidov and S. L. Ziglin, Non-restricted double-averaged three body problem in Hill's case, Celestial Mech., 13 (1976), 471-489. [19] U. Locatelli and A. Giorgilli, Invariant tori in the Sun-Jupiter-Saturn system, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 377-398 (electronic). doi: 10.3934/dcdsb.2007.7.377. [20] F. Malige, P. Robutel and J. Laskar, Partial reduction in the n-body planetary problem using the angular momentum integral, Celestial Mechanics and Dynamical Astronomy, 84 (2002), 283-316. doi: 10.1023/A:1020392219443. [21] R. Moeckel, Some qualitative features of the three-body problem, in "Hamiltonian Dynamical Systems" (Boulder, CO, 1987), Contemp. Math., 81, Amer. Math. Soc., Providence, RI, (1988), 1-22. doi: 10.1090/conm/081/986254. [22] A. Moltchanov, The resonant structure of the solar system, Icarus, 8 (1968), 203-215. [23] J. Moser, "Stable and Random Motions in Dynamical Systems," 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. [24] J. Moser and E. J. Zehnder, "Notes on Dynamical Systems," Courant Lecture Notes in Mathematics, 12, New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2005. [25] N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II, Trudy Sem. Petrovsk., 5 (1979), 5-50. [26] A. Neishtadt, On averaging in two-frequency systems with small Hamiltonian and much smaller non-Hamiltonian perturbations, Dedicated to V. I. Arnold on the occasion of his 65th birthday, Mosc. Math. J., 3 (2003), 1039-1052, 1200. [27] A. Neishtadt, Averaging method and adiabatic invariants, 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. [28] L. Niederman, Stability over exponentially long times in the planetary problem, Nonlinearity, 9 (1996), 1703-1751. doi: 10.1088/0951-7715/9/6/017. [29] G. Pinzari, "On the Kolmogorov Set for Many-Body Problems," Ph.D thesis, Universitá di Roma Tre, 2009. [30] H. Poincaré, "Les Méthodes Nouvelles de la Mécanique Céleste. Tome I, Solutions Périodiques. Non-Existence des Intégrales Uniformes. Solutions Asymptotiques," Librairie Scientifique et Technique Albert Blanchard, Gauthier-Villars, Paris, 1892. [31] H. Poincaré, "Leçons de Mécanique Céleste," Gauthier-Villars, 1905. [32] A. S. Pyartli, Diophantine approximations of submanifolds of a Euclidean space, Funkcional. Anal. i Priložen., 3 (1969), 59-62. [33] P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasiperiodic motions, Celestial Mech. Dynam. Astronom., 62 (1995), 219-261. doi: 10.1007/BF00692089. [34] C. Simó and T. J. Stuchi, Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem, Phys. D, 140 (2000), 1-32. doi: 10.1016/S0167-2789(99)00211-0. [35] E. L. Stiefel and G. Scheifele, "Linear and Regular Celestial Mechanics. Perturbed Two-Body Motion, Numerical Methods, Canonical Theory," Die Grundlehren der mathematischen Wissenschaften, Band 174, Springer-Verlag, New York-Heidelberg, 1971. [36] F. Tisserand, "Traité de Mécanique Céleste," Gauthier-Villars, Paris, 1896.

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##### References:
 [1] K. Abdullah and A. Albouy, On a strange resonance noticed by M. Herman, Regul. Chaotic Dyn., 6 (2001), 421-432. doi: 10.1070/RD2001v006n04ABEH000186. [2] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, 18 (1963), 91-192. [3] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects Of Classical and Celestial Mechanics," Translated from the Russian original by E. Khukhro, Third edition, Encyclopaedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 2006. [4] A. Celletti and L. Chierchia, KAM stability for a three-body problem of the solar system, Z. Angew. Math. Phys., 57 (2006), 33-41. doi: 10.1007/s00033-005-0002-0. [5] A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc., 187 (2007), viii+134 pp. [6] A. Chenciner, Intégration du problème de Kepler par la méthode de Hamilton-Jacobi, Technical report, Bureau des Longitudes, 1989. [7] L. Chierchia and G. Pinzari, Properly-degenerate KAM theory (following V. I. Arnold), Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 545-578. doi: 10.3934/dcdss.2010.3.545. [8] L. Chierchia and G. Pinzari, Planetary Birkhoff normal forms, J. Mod. Dyn., 5 (2011), 623-664. [9] L. Chierchia and G. Pinzari, The planetary $N$-body problem: Symplectic foliation, reductions and invariant tori, Invent. Math., 186 (2011), 1-77. doi: 10.1007/s00222-011-0313-z. [10] J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), Ergodic Theory Dynam. Systems, 24 (2004), 1521-1582. doi: 10.1017/S0143385704000410. [11] J. Féjoz, M. Guàrdia, V. Kaloshin and P. Roldán, Diffusion along mean motion resonance in the restricted planar three-body problem, preprint, 2011, arXiv:1109.2892. [12] J. Galante and V. Kaloshin, Destruction of invariant curves in the restricted circular planar three-body problem by using comparison of action, Duke Math. J., 159 (2011), 275-327. doi: 10.1215/00127094-1415878. [13] M. Hénon, Exploration numérique du problème restreint IV. masses égales, orbites non périodiques, Bulletin Astronomique, 3 (1966), 49-66. [14] A. N. Kolmogorov, On the conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530. [15] J. Laskar, The chaotic motion of the solar system. a numerical estimate of the size of the chaotic zones, Icarus, 88 (1990), 266-291. [16] J. Laskar, Le système solaire est-il stable?, in "Le Chaos," Séminaire Poincaré, XIV, Birkhäuser, (2010), 221-246. [17] J. Laskar and P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian, Celestial Mech. Dynam. Astronom., 62 (1995), 193-217. doi: 10.1007/BF00692088. [18] M. L. Lidov and S. L. Ziglin, Non-restricted double-averaged three body problem in Hill's case, Celestial Mech., 13 (1976), 471-489. [19] U. Locatelli and A. Giorgilli, Invariant tori in the Sun-Jupiter-Saturn system, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 377-398 (electronic). doi: 10.3934/dcdsb.2007.7.377. [20] F. Malige, P. Robutel and J. Laskar, Partial reduction in the n-body planetary problem using the angular momentum integral, Celestial Mechanics and Dynamical Astronomy, 84 (2002), 283-316. doi: 10.1023/A:1020392219443. [21] R. Moeckel, Some qualitative features of the three-body problem, in "Hamiltonian Dynamical Systems" (Boulder, CO, 1987), Contemp. Math., 81, Amer. Math. Soc., Providence, RI, (1988), 1-22. doi: 10.1090/conm/081/986254. [22] A. Moltchanov, The resonant structure of the solar system, Icarus, 8 (1968), 203-215. [23] J. Moser, "Stable and Random Motions in Dynamical Systems," 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. [24] J. Moser and E. J. Zehnder, "Notes on Dynamical Systems," Courant Lecture Notes in Mathematics, 12, New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2005. [25] N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II, Trudy Sem. Petrovsk., 5 (1979), 5-50. [26] A. Neishtadt, On averaging in two-frequency systems with small Hamiltonian and much smaller non-Hamiltonian perturbations, Dedicated to V. I. Arnold on the occasion of his 65th birthday, Mosc. Math. J., 3 (2003), 1039-1052, 1200. [27] A. Neishtadt, Averaging method and adiabatic invariants, 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. [28] L. Niederman, Stability over exponentially long times in the planetary problem, Nonlinearity, 9 (1996), 1703-1751. doi: 10.1088/0951-7715/9/6/017. [29] G. Pinzari, "On the Kolmogorov Set for Many-Body Problems," Ph.D thesis, Universitá di Roma Tre, 2009. [30] H. Poincaré, "Les Méthodes Nouvelles de la Mécanique Céleste. Tome I, Solutions Périodiques. Non-Existence des Intégrales Uniformes. Solutions Asymptotiques," Librairie Scientifique et Technique Albert Blanchard, Gauthier-Villars, Paris, 1892. [31] H. Poincaré, "Leçons de Mécanique Céleste," Gauthier-Villars, 1905. [32] A. S. Pyartli, Diophantine approximations of submanifolds of a Euclidean space, Funkcional. Anal. i Priložen., 3 (1969), 59-62. [33] P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasiperiodic motions, Celestial Mech. Dynam. Astronom., 62 (1995), 219-261. doi: 10.1007/BF00692089. [34] C. Simó and T. J. Stuchi, Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem, Phys. D, 140 (2000), 1-32. doi: 10.1016/S0167-2789(99)00211-0. [35] E. L. Stiefel and G. Scheifele, "Linear and Regular Celestial Mechanics. Perturbed Two-Body Motion, Numerical Methods, Canonical Theory," Die Grundlehren der mathematischen Wissenschaften, Band 174, Springer-Verlag, New York-Heidelberg, 1971. [36] F. Tisserand, "Traité de Mécanique Céleste," Gauthier-Villars, Paris, 1896.
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