American Institute of Mathematical Sciences

Advanced
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.  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.   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.  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.  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.  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.  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.  doi: 10.1017/S0143385708000503.  Google Scholar

[8]

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

[9]

J. Féjoz, Quasiperiodic motions in the planar three-body problem,, J. Differential Equations, 183 (2002), 303.  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.   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, (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.   Google Scholar

[14]

L. Niederman, Stability over exponentially long times in the planetary problem,, Nonlinearity, 9 (1996), 1703.  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.  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.  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.  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.   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.  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.  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.  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.  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.  doi: 10.1017/S0143385708000503.  Google Scholar

[8]

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

[9]

J. Féjoz, Quasiperiodic motions in the planar three-body problem,, J. Differential Equations, 183 (2002), 303.  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.   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, (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.   Google Scholar

[14]

L. Niederman, Stability over exponentially long times in the planetary problem,, Nonlinearity, 9 (1996), 1703.  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.  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.  doi: 10.1023/A:1020355823815.  Google Scholar

[1]

Chiara Caracciolo, Ugo Locatelli. Computer-assisted estimates for Birkhoff normal forms. Journal of Computational Dynamics, 2020, 7 (2) : 425-460. doi: 10.3934/jcd.2020017
[2]

Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313
[3]

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
[4]

Amjad Khan, Dmitry E. Pelinovsky. Long-time stability of small FPU solitary waves. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2065-2075. doi: 10.3934/dcds.2017088
[5]

C. I. Christov, M. D. Todorov. Investigation of the long-time evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139-148. doi: 10.3934/proc.2013.2013.139
[6]

Serge Tabachnikov. Birkhoff billiards are insecure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1035-1040. doi: 10.3934/dcds.2009.23.1035
[7]

Zoltán Buczolich, Balázs Maga, Ryo Moore. Generic Birkhoff spectra. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6649-6679. doi: 10.3934/dcds.2020131
[8]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
[9]

Ke Xu, M. Gregory Forest, Xiaofeng Yang. Shearing the I-N phase transition of liquid crystalline polymers: Long-time memory of defect initial data. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 457-473. doi: 10.3934/dcdsb.2011.15.457
[10]

Linghai Zhang. Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1025-1042. doi: 10.3934/dcds.2002.8.1025
[11]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683
[12]

Pierluigi Colli, Gianni Gilardi, Philippe Laurençot, Amy Novick-Cohen. Uniqueness and long-time behavior for the conserved phase-field system with memory. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 375-390. doi: 10.3934/dcds.1999.5.375
[13]

Tamara Fastovska. Long-time behaviour of a radially symmetric fluid-shell interaction system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1315-1348. doi: 10.3934/dcds.2018054
[14]

Francesca Bucci, Igor Chueshov. Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557
[15]

Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141
[16]

Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure & Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934/cpaa.2007.6.367
[17]

Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379
[18]

Norbert Požár, Giang Thi Thu Vu. Long-time behavior of the one-phase Stefan problem in periodic and random media. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 991-1010. doi: 10.3934/dcdss.2018058
[19]

Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465
[20]

Stefano Galatolo, Mark Holland, Tomas Persson, Yiwei Zhang. Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020341

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences