American Institute of Mathematical Sciences

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  2015, 22: 55-75. doi: 10.3934/era.2015.22.55

Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result

Gabriella Pinzari 1,

1. 

Dipartimento di Matematica ed Applicazioni "R. Caccioppoli”, Università di Napoli "Federico II”, Monte Sant’Angelo – Via Cinthia I-80126 Napoli

Received  June 2014 Revised  February 2015 Published  August 2015

We improve a result in [9] by proving the existence of a positive measure set of $(3n-2)$-dimensional quasi-periodic motions in the spacial, planetary $(1+n)$-body problem away from co-planar, circular motions. We also prove that such quasi-periodic motions reach with continuity corresponding $(2n-1)$-dimensional ones of the planar problem, once the mutual inclinations go to zero (this is related to a speculation in [2]). The main tool is a full reduction of the SO(3)-symmetry, which retains symmetry by reflections and highlights a quasi-integrable structure, with a small remainder, independently of eccentricities and inclinations.
Keywords: Quasi-integrable structures for perturbed super-integrable systems, Deprit's reduction, perihelia reduction, Symmetries, $N$-body problem, Multi-scale KAM Theory., canonical coordinates, Arnold's Theorem on the stability of planetary motions, Jacobi reduction.
Mathematics Subject Classification: 34D10, 34C20, 70E55, 70F10, 70F15, 70F07, 37J10, 37J15, 37J25, 37J35, 37J40, 70K4.
Citation: Gabriella Pinzari. Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result. Electronic Research Announcements, 2015, 22: 55-75. doi: 10.3934/era.2015.22.55
References:
[1]

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

[2]

V. I. Arnol'd, Small denominators and problems of stability of motion in classical and celestial mechanics,, \emph{Russian Math. Surveys}, 18 (1963), 85.  doi: 10.1070/RM1963v018n06ABEH001143.  Google Scholar

[3]

F. Boigey, Élimination des nœ uds dans le problème newtonien des quatre corps,, \emph{Celestial Mech.}, 27 (1982), 399.  doi: 10.1007/BF01228562.  Google Scholar

[4]

L. Chierchia, The Planetary N-Body Problem,, \emph{UNESCO Encyclopedia of Life Support Systems}, (2012).   Google Scholar

[5]

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

[6]

L. Chierchia and G. Pinzari, Deprit's reduction of the nodes revised,, \emph{Celestial Mech. Dynam. Astronom.}, 109 (2011), 285.  doi: 10.1007/s10569-010-9329-8.  Google Scholar

[7]

L. Chierchia and G. Pinzari, Metric stability of the planetary N-body problem,, \emph{Proceedings of the International Congress of Mathematicians}, (2014).   Google Scholar

[8]

L. Chierchia and G. Pinzari, Planetary Birkhoff normal forms,, \emph{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,, \emph{Invent. Math.}, 186 (2011), 1.  doi: 10.1007/s00222-011-0313-z.  Google Scholar

[10]

A. Deprit, Elimination of the nodes in problems of $n$ bodies,, \emph{Celestial Mech.}, 30 (1983), 181.  doi: 10.1007/BF01234305.  Google Scholar

[11]

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

[12]

J. Féjoz, On "Arnold's theorem'' on the stability of the solar system,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 3555.  doi: 10.3934/dcds.2013.33.3555.  Google Scholar

[13]

J. Féjoz, On action-angle coordinates and the Poincaré coordinates,, \emph{Regul. Chaotic Dyn.}, 18 (2013), 703.  doi: 10.1134/S1560354713060105.  Google Scholar

[14]

S. Ferrer and C. Osácar, Harrington's Hamiltonian in the stellar problem of three bodies: Reductions, relative equilibria and bifurcations,, \emph{Celestial Mech. Dynam. Astronom.}, 58 (1994), 245.  doi: 10.1007/BF00691977.  Google Scholar

[15]

R. S. Harrington, The stellar three-body problem,, \emph{Celestial Mech. and Dyn. Astrronom}, 1 (1969), 200.  doi: 10.1007/BF01228839.  Google Scholar

[16]

M. R. Herman, Torsion du problème planétaire, edited by J. Féjoz, 2009., Available electronically at \url{http://www.college-de-france.fr/media/jean-christophe-yoccoz/UPL61526_FonctionPerturbatrice_2009_02.pdf}., ().   Google Scholar

[17]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics,, Modern Birkhäuser Classics, (1994).  doi: 10.1007/978-3-0348-0104-1.  Google Scholar

[18]

C. G. J. Jacobi, Sur l'élimination des noeuds dans le problème des trois corps,, \emph{Astronomische Nachrichten}, 20 (1843), 81.  doi: 10.1002/asna.18430200602.  Google Scholar

[19]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function,, \emph{Dokl. Akad. Nauk SSSR (N. S.)}, 98 (1954), 527.   Google Scholar

[20]

J. Laskar and P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian,, \emph{Celestial Mech. Dynam. Astronom.}, 62 (1995), 193.  doi: 10.1007/BF00692088.  Google Scholar

[21]

M. L. Lidov and S. L. Ziglin, Non-restricted double-averaged three body problem in Hill's case,, \emph{Celestial Mech.}, 13 (1976), 471.  doi: 10.1007/BF01229100.  Google Scholar

[22]

F. Malige, P. Robutel and J. Laskar, Partial reduction in the $n$-body planetary problem using the angular momentum integral,, \emph{Celestial Mech. Dynam. Astronom.}, 84 (2002), 283.  doi: 10.1023/A:1020392219443.  Google Scholar

[23]

G. Pinzari, On the Kolmogorov set for many-body problems,, Ph.D thesis, (2009).   Google Scholar

[24]

G. Pinzari, Aspects of the planetary Birkhoff normal form,, \emph{Regul. Chaotic Dyn.}, 18 (2013), 860.  doi: 10.1134/S1560354713060178.  Google Scholar

[25]

G. Pinzari, Perihelia reduction and global Kolmogorov tori in the planetary problem,, \arXiv{1501.04470}., ().   Google Scholar

[26]

J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems,, \emph{Math. Z.}, 213 (1993), 187.  doi: 10.1007/BF03025718.  Google Scholar

[27]

R. Radau, Sur une transformation des équations différentielles de la dynamique,, \emph{Ann. Sci. Éc. Norm. Sup.}, 5 (1868), 311.   Google Scholar

[28]

P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasiperiodic motions,, \emph{Celestial Mech. Dynam. Astronom.}, 62 (1995), 219.  doi: 10.1007/BF00692089.  Google Scholar

[29]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems,, \emph{Regul. Chaotic Dyn.}, 6 (2001), 119.  doi: 10.1070/RD2001v006n02ABEH000169.  Google Scholar

show all references

References:
[1]

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

[2]

V. I. Arnol'd, Small denominators and problems of stability of motion in classical and celestial mechanics,, \emph{Russian Math. Surveys}, 18 (1963), 85.  doi: 10.1070/RM1963v018n06ABEH001143.  Google Scholar

[3]

F. Boigey, Élimination des nœ uds dans le problème newtonien des quatre corps,, \emph{Celestial Mech.}, 27 (1982), 399.  doi: 10.1007/BF01228562.  Google Scholar

[4]

L. Chierchia, The Planetary N-Body Problem,, \emph{UNESCO Encyclopedia of Life Support Systems}, (2012).   Google Scholar

[5]

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

[6]

L. Chierchia and G. Pinzari, Deprit's reduction of the nodes revised,, \emph{Celestial Mech. Dynam. Astronom.}, 109 (2011), 285.  doi: 10.1007/s10569-010-9329-8.  Google Scholar

[7]

L. Chierchia and G. Pinzari, Metric stability of the planetary N-body problem,, \emph{Proceedings of the International Congress of Mathematicians}, (2014).   Google Scholar

[8]

L. Chierchia and G. Pinzari, Planetary Birkhoff normal forms,, \emph{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,, \emph{Invent. Math.}, 186 (2011), 1.  doi: 10.1007/s00222-011-0313-z.  Google Scholar

[10]

A. Deprit, Elimination of the nodes in problems of $n$ bodies,, \emph{Celestial Mech.}, 30 (1983), 181.  doi: 10.1007/BF01234305.  Google Scholar

[11]

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

[12]

J. Féjoz, On "Arnold's theorem'' on the stability of the solar system,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 3555.  doi: 10.3934/dcds.2013.33.3555.  Google Scholar

[13]

J. Féjoz, On action-angle coordinates and the Poincaré coordinates,, \emph{Regul. Chaotic Dyn.}, 18 (2013), 703.  doi: 10.1134/S1560354713060105.  Google Scholar

[14]

S. Ferrer and C. Osácar, Harrington's Hamiltonian in the stellar problem of three bodies: Reductions, relative equilibria and bifurcations,, \emph{Celestial Mech. Dynam. Astronom.}, 58 (1994), 245.  doi: 10.1007/BF00691977.  Google Scholar

[15]

R. S. Harrington, The stellar three-body problem,, \emph{Celestial Mech. and Dyn. Astrronom}, 1 (1969), 200.  doi: 10.1007/BF01228839.  Google Scholar

[16]

M. R. Herman, Torsion du problème planétaire, edited by J. Féjoz, 2009., Available electronically at \url{http://www.college-de-france.fr/media/jean-christophe-yoccoz/UPL61526_FonctionPerturbatrice_2009_02.pdf}., ().   Google Scholar

[17]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics,, Modern Birkhäuser Classics, (1994).  doi: 10.1007/978-3-0348-0104-1.  Google Scholar

[18]

C. G. J. Jacobi, Sur l'élimination des noeuds dans le problème des trois corps,, \emph{Astronomische Nachrichten}, 20 (1843), 81.  doi: 10.1002/asna.18430200602.  Google Scholar

[19]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function,, \emph{Dokl. Akad. Nauk SSSR (N. S.)}, 98 (1954), 527.   Google Scholar

[20]

J. Laskar and P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian,, \emph{Celestial Mech. Dynam. Astronom.}, 62 (1995), 193.  doi: 10.1007/BF00692088.  Google Scholar

[21]

M. L. Lidov and S. L. Ziglin, Non-restricted double-averaged three body problem in Hill's case,, \emph{Celestial Mech.}, 13 (1976), 471.  doi: 10.1007/BF01229100.  Google Scholar

[22]

F. Malige, P. Robutel and J. Laskar, Partial reduction in the $n$-body planetary problem using the angular momentum integral,, \emph{Celestial Mech. Dynam. Astronom.}, 84 (2002), 283.  doi: 10.1023/A:1020392219443.  Google Scholar

[23]

G. Pinzari, On the Kolmogorov set for many-body problems,, Ph.D thesis, (2009).   Google Scholar

[24]

G. Pinzari, Aspects of the planetary Birkhoff normal form,, \emph{Regul. Chaotic Dyn.}, 18 (2013), 860.  doi: 10.1134/S1560354713060178.  Google Scholar

[25]

G. Pinzari, Perihelia reduction and global Kolmogorov tori in the planetary problem,, \arXiv{1501.04470}., ().   Google Scholar

[26]

J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems,, \emph{Math. Z.}, 213 (1993), 187.  doi: 10.1007/BF03025718.  Google Scholar

[27]

R. Radau, Sur une transformation des équations différentielles de la dynamique,, \emph{Ann. Sci. Éc. Norm. Sup.}, 5 (1868), 311.   Google Scholar

[28]

P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasiperiodic motions,, \emph{Celestial Mech. Dynam. Astronom.}, 62 (1995), 219.  doi: 10.1007/BF00692089.  Google Scholar

[29]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems,, \emph{Regul. Chaotic Dyn.}, 6 (2001), 119.  doi: 10.1070/RD2001v006n02ABEH000169.  Google Scholar

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