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October  2011, 5(4): 623-664. doi: 10.3934/jmd.2011.5.623

Planetary Birkhoff normal forms

Luigi Chierchia 1, and  Gabriella Pinzari 2,

1. 

Dipartimento di Matematica, Università "Roma Tre", Largo S. L. Murialdo 1, 00146 Roma
2. 

Dipartimento di Matematica, Università “Roma Tre”, Largo S. L. Murialdo 1, I-00146 Roma, Italy

Received  September 2010 Revised  October 2011 Published  March 2012

Birkhoff normal forms for the (secular) planetary problem are investigated. Existence and uniqueness is discussed and it is shown that the classical Poincaré variables and the ʀᴘs-variables (introduced in [6]), after a trivial lift, lead to the same Birkhoff normal form; as a corollary the Birkhoff normal form (in Poincaré variables) is degenerate at all orders (answering a question of M. Herman). Non-degenerate Birkhoff normal forms for partially and totally reduced cases are provided and an application to long-time stability of secular action variables (eccentricities and inclinations) is discussed.
Keywords: Birkhoff normal form, Planetary system, N-body problem, Birkhoff invariants, Long-time stability..
Mathematics Subject Classification: 70F10, 70K45, 70F15, 70F07, 70E55, 34C20, 34D1.
Citation: Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623
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, Uspehi Mat. Nauk, 18 (1963), 91-192.  Google Scholar

[3]

L. Biasco, L. Chierchia and E. Valdinoci, Elliptic two-dimensional invariant tori for the planetary three-body problem, Arch. Rational Mech. Anal., 170 (2003), 91-135. See also Corrigendum, Arch. Ration. Mech. Anal., 180 (2006), 507-509. doi: 10.1007/s00205-005-0410-5.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

L. Chierchia and F. Pusateri, Analytic Lagrangian tori for the planetary many-body problem, Ergodic Theory Dynam. Systems, 29 (2009), 849-873. doi: 10.1017/S0143385708000503.  Google Scholar

[8]

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

[9]

J. Féjoz, Quasiperiodic motions in the planar three-body problem, J. Differential Equations, 183 (2002), 303-341. doi: 10.1006/jdeq.2001.4117.  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), 1521-1582.  Google Scholar

[11]

M. R. Herman, Torsion du problème planètaire, ed. J. Fejóz, 'Archives Michel Herman', 2009., Available from: \url{http://www.college-de-france.fr/default/EN/all/equ_dif/archives_michel_herman.htm}., ().   Google Scholar

[12]

H. Hofer and E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics," Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar

[13]

N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5-66, 287.  Google Scholar

[14]

L. Niederman, Stability over exponentially long times in the planetary problem, Nonlinearity, 9 (1996), 1703-1751. doi: 10.1088/0951-7715/9/6/017.  Google Scholar

[15]

J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[16]

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. See also Erratum, Celestial Mech. Dynam. Astronom., 84 (2002), 317. doi: 10.1023/A:1020355823815.  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, Uspehi Mat. Nauk, 18 (1963), 91-192.  Google Scholar

[3]

L. Biasco, L. Chierchia and E. Valdinoci, Elliptic two-dimensional invariant tori for the planetary three-body problem, Arch. Rational Mech. Anal., 170 (2003), 91-135. See also Corrigendum, Arch. Ration. Mech. Anal., 180 (2006), 507-509. doi: 10.1007/s00205-005-0410-5.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

L. Chierchia and F. Pusateri, Analytic Lagrangian tori for the planetary many-body problem, Ergodic Theory Dynam. Systems, 29 (2009), 849-873. doi: 10.1017/S0143385708000503.  Google Scholar

[8]

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

[9]

J. Féjoz, Quasiperiodic motions in the planar three-body problem, J. Differential Equations, 183 (2002), 303-341. doi: 10.1006/jdeq.2001.4117.  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), 1521-1582.  Google Scholar

[11]

M. R. Herman, Torsion du problème planètaire, ed. J. Fejóz, 'Archives Michel Herman', 2009., Available from: \url{http://www.college-de-france.fr/default/EN/all/equ_dif/archives_michel_herman.htm}., ().   Google Scholar

[12]

H. Hofer and E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics," Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar

[13]

N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5-66, 287.  Google Scholar

[14]

L. Niederman, Stability over exponentially long times in the planetary problem, Nonlinearity, 9 (1996), 1703-1751. doi: 10.1088/0951-7715/9/6/017.  Google Scholar

[15]

J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.  Google Scholar

[16]

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. See also Erratum, Celestial Mech. Dynam. Astronom., 84 (2002), 317. doi: 10.1023/A:1020355823815.  Google Scholar

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