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Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result
1. | Dipartimento di Matematica ed Applicazioni "R. Caccioppoli”, Università di Napoli "Federico II”, Monte Sant’Angelo – Via Cinthia I-80126 Napoli |
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. 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. |
[3] |
F. Boigey, Élimination des nœ uds dans le problème newtonien des quatre corps, Celestial Mech., 27 (1982), 399-414.
doi: 10.1007/BF01228562. |
[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. |
[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. |
[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. |
[7] |
L. Chierchia and G. Pinzari, Metric stability of the planetary N-body problem, Proceedings of the International Congress of Mathematicians, 2014. |
[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] |
A. Deprit, Elimination of the nodes in problems of $n$ bodies, Celestial Mech., 30 (1983), 181-195.
doi: 10.1007/BF01234305. |
[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. |
[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. |
[13] |
J. Féjoz, On action-angle coordinates and the Poincaré coordinates, Regul. Chaotic Dyn., 18 (2013), 703-718.
doi: 10.1134/S1560354713060105. |
[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. |
[15] |
R. S. Harrington, The stellar three-body problem, Celestial Mech. and Dyn. Astrronom, 1 (1969), 200-209.
doi: 10.1007/BF01228839. |
[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}., ().
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
G. Pinzari, On the Kolmogorov set for many-body problems, Ph.D thesis, Università Roma Tre, April 2009. |
[24] |
G. Pinzari, Aspects of the planetary Birkhoff normal form, Regul. Chaotic Dyn., 18 (2013), 860-906.
doi: 10.1134/S1560354713060178. |
[25] |
G. Pinzari, Perihelia reduction and global Kolmogorov tori in the planetary problem,, \arXiv{1501.04470}., ().
|
[26] |
J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216.
doi: 10.1007/BF03025718. |
[27] |
R. Radau, Sur une transformation des équations différentielles de la dynamique, Ann. Sci. Éc. Norm. Sup., 5 (1868), 311-375. |
[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. |
[29] |
H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204.
doi: 10.1070/RD2001v006n02ABEH000169. |
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. |
[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. |
[3] |
F. Boigey, Élimination des nœ uds dans le problème newtonien des quatre corps, Celestial Mech., 27 (1982), 399-414.
doi: 10.1007/BF01228562. |
[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. |
[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. |
[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. |
[7] |
L. Chierchia and G. Pinzari, Metric stability of the planetary N-body problem, Proceedings of the International Congress of Mathematicians, 2014. |
[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] |
A. Deprit, Elimination of the nodes in problems of $n$ bodies, Celestial Mech., 30 (1983), 181-195.
doi: 10.1007/BF01234305. |
[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. |
[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. |
[13] |
J. Féjoz, On action-angle coordinates and the Poincaré coordinates, Regul. Chaotic Dyn., 18 (2013), 703-718.
doi: 10.1134/S1560354713060105. |
[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. |
[15] |
R. S. Harrington, The stellar three-body problem, Celestial Mech. and Dyn. Astrronom, 1 (1969), 200-209.
doi: 10.1007/BF01228839. |
[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}., ().
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
G. Pinzari, On the Kolmogorov set for many-body problems, Ph.D thesis, Università Roma Tre, April 2009. |
[24] |
G. Pinzari, Aspects of the planetary Birkhoff normal form, Regul. Chaotic Dyn., 18 (2013), 860-906.
doi: 10.1134/S1560354713060178. |
[25] |
G. Pinzari, Perihelia reduction and global Kolmogorov tori in the planetary problem,, \arXiv{1501.04470}., ().
|
[26] |
J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216.
doi: 10.1007/BF03025718. |
[27] |
R. Radau, Sur une transformation des équations différentielles de la dynamique, Ann. Sci. Éc. Norm. Sup., 5 (1868), 311-375. |
[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. |
[29] |
H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204.
doi: 10.1070/RD2001v006n02ABEH000169. |
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