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 - A, 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.  doi: 10.1070/RD2001v006n04ABEH000186.  Google Scholar

[2]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics,, Uspehi Mat. Nauk, 18 (1963), 91.   Google Scholar

[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, (2006).   Google Scholar

[4]

A. Celletti and L. Chierchia, KAM stability for a three-body problem of the solar system,, Z. Angew. Math. Phys., 57 (2006), 33.  doi: 10.1007/s00033-005-0002-0.  Google Scholar

[5]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics,, Mem. Amer. Math. Soc., 187 (2007).   Google Scholar

[6]

A. Chenciner, Intégration du problème de Kepler par la méthode de Hamilton-Jacobi,, Technical report, (1989).   Google Scholar

[7]

L. Chierchia and G. Pinzari, Properly-degenerate KAM theory (following V. I. Arnold),, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 545.  doi: 10.3934/dcdss.2010.3.545.  Google Scholar

[8]

L. Chierchia and G. Pinzari, Planetary Birkhoff normal forms,, J. Mod. Dyn., 5 (2011), 623.   Google Scholar

[9]

L. Chierchia and G. Pinzari, The planetary $N$-body problem: Symplectic foliation, reductions and invariant tori,, Invent. Math., 186 (2011), 1.  doi: 10.1007/s00222-011-0313-z.  Google Scholar

[10]

J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman),, Ergodic Theory Dynam. Systems, 24 (2004), 1521.  doi: 10.1017/S0143385704000410.  Google Scholar

[11]

J. Féjoz, M. Guàrdia, V. Kaloshin and P. Roldán, Diffusion along mean motion resonance in the restricted planar three-body problem,, preprint, (2011).   Google Scholar

[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.  doi: 10.1215/00127094-1415878.  Google Scholar

[13]

M. Hénon, Exploration numérique du problème restreint IV. masses égales, orbites non périodiques,, Bulletin Astronomique, 3 (1966), 49.   Google Scholar

[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.   Google Scholar

[15]

J. Laskar, The chaotic motion of the solar system. a numerical estimate of the size of the chaotic zones,, Icarus, 88 (1990), 266.   Google Scholar

[16]

J. Laskar, Le système solaire est-il stable?,, in, XIV (2010), 221.   Google Scholar

[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.  doi: 10.1007/BF00692088.  Google Scholar

[18]

M. L. Lidov and S. L. Ziglin, Non-restricted double-averaged three body problem in Hill's case,, Celestial Mech., 13 (1976), 471.   Google Scholar

[19]

U. Locatelli and A. Giorgilli, Invariant tori in the Sun-Jupiter-Saturn system,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 377.  doi: 10.3934/dcdsb.2007.7.377.  Google Scholar

[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.  doi: 10.1023/A:1020392219443.  Google Scholar

[21]

R. Moeckel, Some qualitative features of the three-body problem,, in, 81 (1988), 1.  doi: 10.1090/conm/081/986254.  Google Scholar

[22]

A. Moltchanov, The resonant structure of the solar system,, Icarus, 8 (1968), 203.   Google Scholar

[23]

J. Moser, "Stable and Random Motions in Dynamical Systems,", With special emphasis on celestial mechanics, (1973).   Google Scholar

[24]

J. Moser and E. J. Zehnder, "Notes on Dynamical Systems," Courant Lecture Notes in Mathematics, 12,, New York University, (2005).   Google Scholar

[25]

N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II,, Trudy Sem. Petrovsk., 5 (1979), 5.   Google Scholar

[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, 3 (2003), 1039.   Google Scholar

[27]

A. Neishtadt, Averaging method and adiabatic invariants,, in, (2008), 53.  doi: 10.1007/978-1-4020-6964-2_3.  Google Scholar

[28]

L. Niederman, Stability over exponentially long times in the planetary problem,, Nonlinearity, 9 (1996), 1703.  doi: 10.1088/0951-7715/9/6/017.  Google Scholar

[29]

G. Pinzari, "On the Kolmogorov Set for Many-Body Problems,", Ph.D thesis, (2009).   Google Scholar

[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, (1892).   Google Scholar

[31]

H. Poincaré, "Leçons de Mécanique Céleste,", Gauthier-Villars, (1905).   Google Scholar

[32]

A. S. Pyartli, Diophantine approximations of submanifolds of a Euclidean space,, Funkcional. Anal. i Priložen., 3 (1969), 59.   Google Scholar

[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.  doi: 10.1007/BF00692089.  Google Scholar

[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.  doi: 10.1016/S0167-2789(99)00211-0.  Google Scholar

[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, (1971).   Google Scholar

[36]

F. Tisserand, "Traité de Mécanique Céleste,", Gauthier-Villars, (1896).   Google Scholar

show all references

References:
[1]

K. Abdullah and A. Albouy, On a strange resonance noticed by M. Herman,, Regul. Chaotic Dyn., 6 (2001), 421.  doi: 10.1070/RD2001v006n04ABEH000186.  Google Scholar

[2]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics,, Uspehi Mat. Nauk, 18 (1963), 91.   Google Scholar

[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, (2006).   Google Scholar

[4]

A. Celletti and L. Chierchia, KAM stability for a three-body problem of the solar system,, Z. Angew. Math. Phys., 57 (2006), 33.  doi: 10.1007/s00033-005-0002-0.  Google Scholar

[5]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics,, Mem. Amer. Math. Soc., 187 (2007).   Google Scholar

[6]

A. Chenciner, Intégration du problème de Kepler par la méthode de Hamilton-Jacobi,, Technical report, (1989).   Google Scholar

[7]

L. Chierchia and G. Pinzari, Properly-degenerate KAM theory (following V. I. Arnold),, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 545.  doi: 10.3934/dcdss.2010.3.545.  Google Scholar

[8]

L. Chierchia and G. Pinzari, Planetary Birkhoff normal forms,, J. Mod. Dyn., 5 (2011), 623.   Google Scholar

[9]

L. Chierchia and G. Pinzari, The planetary $N$-body problem: Symplectic foliation, reductions and invariant tori,, Invent. Math., 186 (2011), 1.  doi: 10.1007/s00222-011-0313-z.  Google Scholar

[10]

J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman),, Ergodic Theory Dynam. Systems, 24 (2004), 1521.  doi: 10.1017/S0143385704000410.  Google Scholar

[11]

J. Féjoz, M. Guàrdia, V. Kaloshin and P. Roldán, Diffusion along mean motion resonance in the restricted planar three-body problem,, preprint, (2011).   Google Scholar

[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.  doi: 10.1215/00127094-1415878.  Google Scholar

[13]

M. Hénon, Exploration numérique du problème restreint IV. masses égales, orbites non périodiques,, Bulletin Astronomique, 3 (1966), 49.   Google Scholar

[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.   Google Scholar

[15]

J. Laskar, The chaotic motion of the solar system. a numerical estimate of the size of the chaotic zones,, Icarus, 88 (1990), 266.   Google Scholar

[16]

J. Laskar, Le système solaire est-il stable?,, in, XIV (2010), 221.   Google Scholar

[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.  doi: 10.1007/BF00692088.  Google Scholar

[18]

M. L. Lidov and S. L. Ziglin, Non-restricted double-averaged three body problem in Hill's case,, Celestial Mech., 13 (1976), 471.   Google Scholar

[19]

U. Locatelli and A. Giorgilli, Invariant tori in the Sun-Jupiter-Saturn system,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 377.  doi: 10.3934/dcdsb.2007.7.377.  Google Scholar

[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.  doi: 10.1023/A:1020392219443.  Google Scholar

[21]

R. Moeckel, Some qualitative features of the three-body problem,, in, 81 (1988), 1.  doi: 10.1090/conm/081/986254.  Google Scholar

[22]

A. Moltchanov, The resonant structure of the solar system,, Icarus, 8 (1968), 203.   Google Scholar

[23]

J. Moser, "Stable and Random Motions in Dynamical Systems,", With special emphasis on celestial mechanics, (1973).   Google Scholar

[24]

J. Moser and E. J. Zehnder, "Notes on Dynamical Systems," Courant Lecture Notes in Mathematics, 12,, New York University, (2005).   Google Scholar

[25]

N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II,, Trudy Sem. Petrovsk., 5 (1979), 5.   Google Scholar

[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, 3 (2003), 1039.   Google Scholar

[27]

A. Neishtadt, Averaging method and adiabatic invariants,, in, (2008), 53.  doi: 10.1007/978-1-4020-6964-2_3.  Google Scholar

[28]

L. Niederman, Stability over exponentially long times in the planetary problem,, Nonlinearity, 9 (1996), 1703.  doi: 10.1088/0951-7715/9/6/017.  Google Scholar

[29]

G. Pinzari, "On the Kolmogorov Set for Many-Body Problems,", Ph.D thesis, (2009).   Google Scholar

[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, (1892).   Google Scholar

[31]

H. Poincaré, "Leçons de Mécanique Céleste,", Gauthier-Villars, (1905).   Google Scholar

[32]

A. S. Pyartli, Diophantine approximations of submanifolds of a Euclidean space,, Funkcional. Anal. i Priložen., 3 (1969), 59.   Google Scholar

[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.  doi: 10.1007/BF00692089.  Google Scholar

[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.  doi: 10.1016/S0167-2789(99)00211-0.  Google Scholar

[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, (1971).   Google Scholar

[36]

F. Tisserand, "Traité de Mécanique Céleste,", Gauthier-Villars, (1896).   Google Scholar

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