Planetary Birkhoff normal forms

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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

Abstract
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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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	A. Deprit, Elimination of the nodes in problems of $n$ bodies, Celestial Mech., 30 (1983), 181-195. doi: 10.1007/BF01234305.
[9]	J. Féjoz, Quasiperiodic motions in the planar three-body problem, J. Differential Equations, 183 (2002), 303-341. doi: 10.1006/jdeq.2001.4117.
[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.
[11]	M. R. Herman, Torsion du problème planètaire, ed. J. Fejóz, 'Archives Michel Herman', 2009. Available from: http://www.college-de-france.fr/default/EN/all/equ_dif/archives_michel_herman.htm.
[12]	H. Hofer and E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics," Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.
[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.
[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.
[15]	J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.
[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.

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, Uspehi Mat. Nauk, 18 (1963), 91-192.
[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.
[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.
[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.
[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.
[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.
[8]	A. Deprit, Elimination of the nodes in problems of $n$ bodies, Celestial Mech., 30 (1983), 181-195. doi: 10.1007/BF01234305.
[9]	J. Féjoz, Quasiperiodic motions in the planar three-body problem, J. Differential Equations, 183 (2002), 303-341. doi: 10.1006/jdeq.2001.4117.
[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.
[11]	M. R. Herman, Torsion du problème planètaire, ed. J. Fejóz, 'Archives Michel Herman', 2009. Available from: http://www.college-de-france.fr/default/EN/all/equ_dif/archives_michel_herman.htm.
[12]	H. Hofer and E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics," Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.
[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.
[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.
[15]	J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.
[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.

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