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

[2]

V. I. Arnol'd, Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys, 18 (1963), 85-191. doi: 10.1070/RM1963v018n06ABEH001143.  Google Scholar

[3]

F. Boigey, Élimination des nœ uds dans le problème newtonien des quatre corps, Celestial Mech., 27 (1982), 399-414. doi: 10.1007/BF01228562.  Google Scholar

[4]

L. Chierchia, The Planetary N-Body Problem, UNESCO Encyclopedia of Life Support Systems, Vol. 6.119.55, Celestial Mechanics, Eolss Publishers Co Ltd, 2012. Google Scholar

[5]

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.  Google Scholar

[6]

L. Chierchia and G. Pinzari, Deprit's reduction of the nodes revised, Celestial Mech. Dynam. Astronom., 109 (2011), 285-301. doi: 10.1007/s10569-010-9329-8.  Google Scholar

[7]

L. Chierchia and G. Pinzari, Metric stability of the planetary N-body problem, Proceedings of the International Congress of Mathematicians, 2014. Google Scholar

[8]

L. Chierchia and G. Pinzari, Planetary Birkhoff normal forms, J. Mod. Dyn., 5 (2011), 623-664.  Google Scholar

[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.  Google Scholar

[10]

A. Deprit, Elimination of the nodes in problems of $n$ bodies, Celestial Mech., 30 (1983), 181-195. doi: 10.1007/BF01234305.  Google Scholar

[11]

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.  Google Scholar

[12]

J. Féjoz, On "Arnold's theorem'' on the stability of the solar system, Discrete Contin. Dyn. Syst., 33 (2013), 3555-3565. doi: 10.3934/dcds.2013.33.3555.  Google Scholar

[13]

J. Féjoz, On action-angle coordinates and the Poincaré coordinates, Regul. Chaotic Dyn., 18 (2013), 703-718. 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, Celestial Mech. Dynam. Astronom., 58 (1994), 245-275. doi: 10.1007/BF00691977.  Google Scholar

[15]

R. S. Harrington, The stellar three-body problem, Celestial Mech. and Dyn. Astrronom, 1 (1969), 200-209. 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, Birkhäuser, Basel, 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, Astronomische Nachrichten, 20 (1843), 81-98. 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, Dokl. Akad. Nauk SSSR (N. S.), 98 (1954), 527-530.  Google Scholar

[20]

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.  Google Scholar

[21]

M. L. Lidov and S. L. Ziglin, Non-restricted double-averaged three body problem in Hill's case, Celestial Mech., 13 (1976), 471-489. 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, Celestial Mech. Dynam. Astronom., 84 (2002), 283-316. doi: 10.1023/A:1020392219443.  Google Scholar

[23]

G. Pinzari, On the Kolmogorov set for many-body problems, Ph.D thesis, Università Roma Tre, April 2009. Google Scholar

[24]

G. Pinzari, Aspects of the planetary Birkhoff normal form, Regul. Chaotic Dyn., 18 (2013), 860-906. 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, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[27]

R. Radau, Sur une transformation des équations différentielles de la dynamique, Ann. Sci. Éc. Norm. Sup., 5 (1868), 311-375.  Google Scholar

[28]

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.  Google Scholar

[29]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204. doi: 10.1070/RD2001v006n02ABEH000169.  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), 421-432. doi: 10.1070/RD2001v006n04ABEH000186.  Google Scholar

[2]

V. I. Arnol'd, Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys, 18 (1963), 85-191. doi: 10.1070/RM1963v018n06ABEH001143.  Google Scholar

[3]

F. Boigey, Élimination des nœ uds dans le problème newtonien des quatre corps, Celestial Mech., 27 (1982), 399-414. doi: 10.1007/BF01228562.  Google Scholar

[4]

L. Chierchia, The Planetary N-Body Problem, UNESCO Encyclopedia of Life Support Systems, Vol. 6.119.55, Celestial Mechanics, Eolss Publishers Co Ltd, 2012. Google Scholar

[5]

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.  Google Scholar

[6]

L. Chierchia and G. Pinzari, Deprit's reduction of the nodes revised, Celestial Mech. Dynam. Astronom., 109 (2011), 285-301. doi: 10.1007/s10569-010-9329-8.  Google Scholar

[7]

L. Chierchia and G. Pinzari, Metric stability of the planetary N-body problem, Proceedings of the International Congress of Mathematicians, 2014. Google Scholar

[8]

L. Chierchia and G. Pinzari, Planetary Birkhoff normal forms, J. Mod. Dyn., 5 (2011), 623-664.  Google Scholar

[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.  Google Scholar

[10]

A. Deprit, Elimination of the nodes in problems of $n$ bodies, Celestial Mech., 30 (1983), 181-195. doi: 10.1007/BF01234305.  Google Scholar

[11]

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.  Google Scholar

[12]

J. Féjoz, On "Arnold's theorem'' on the stability of the solar system, Discrete Contin. Dyn. Syst., 33 (2013), 3555-3565. doi: 10.3934/dcds.2013.33.3555.  Google Scholar

[13]

J. Féjoz, On action-angle coordinates and the Poincaré coordinates, Regul. Chaotic Dyn., 18 (2013), 703-718. 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, Celestial Mech. Dynam. Astronom., 58 (1994), 245-275. doi: 10.1007/BF00691977.  Google Scholar

[15]

R. S. Harrington, The stellar three-body problem, Celestial Mech. and Dyn. Astrronom, 1 (1969), 200-209. 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, Birkhäuser, Basel, 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, Astronomische Nachrichten, 20 (1843), 81-98. 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, Dokl. Akad. Nauk SSSR (N. S.), 98 (1954), 527-530.  Google Scholar

[20]

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.  Google Scholar

[21]

M. L. Lidov and S. L. Ziglin, Non-restricted double-averaged three body problem in Hill's case, Celestial Mech., 13 (1976), 471-489. 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, Celestial Mech. Dynam. Astronom., 84 (2002), 283-316. doi: 10.1023/A:1020392219443.  Google Scholar

[23]

G. Pinzari, On the Kolmogorov set for many-body problems, Ph.D thesis, Università Roma Tre, April 2009. Google Scholar

[24]

G. Pinzari, Aspects of the planetary Birkhoff normal form, Regul. Chaotic Dyn., 18 (2013), 860-906. 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, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[27]

R. Radau, Sur une transformation des équations différentielles de la dynamique, Ann. Sci. Éc. Norm. Sup., 5 (1868), 311-375.  Google Scholar

[28]

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.  Google Scholar

[29]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204. doi: 10.1070/RD2001v006n02ABEH000169.  Google Scholar

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