American Institute of Mathematical Sciences

Advanced
March  2007, 7(2): 377-398. doi: 10.3934/dcdsb.2007.7.377

Invariant tori in the Sun--Jupiter--Saturn system

Ugo Locatelli 1, and  Antonio Giorgilli 2,

1. 

Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma, Italy
2. 

Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy

Received  January 2006 Revised  November 2006 Published  December 2006

We discuss the applicability of Kolmogorov's theorem on existence of invariant tori to the real Sun-Jupiter-Saturn system. Using computer algebra, we construct a Kolmogorov's normal form defined in a neighborhood of the actual orbit in the phase space, giving a sharp evidence of the convergence of the algorithm. If not a rigorous proof, we consider our calculation as a strong indication that Kolmogorov's theorem applies to the motion of the two biggest planets of our solar system.
Keywords: Celestial echanics., Hamiltonian systems, Three--body problem, perturbation methods, KAM theory.
Mathematics Subject Classification: Primary: 70F07; Secondary: 70F10, 37J40, 37N05, 70H0.
Citation: Ugo Locatelli, Antonio Giorgilli. Invariant tori in the Sun--Jupiter--Saturn system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 377-398. doi: 10.3934/dcdsb.2007.7.377
[1]

Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533
[2]

Marian Gidea, Rafael De La Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2006, 14 (2) : 295-328. doi: 10.3934/dcds.2006.14.295
[3]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623
[4]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463
[5]

Edward Belbruno. Random walk in the three-body problem and applications. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519
[6]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684
[7]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[8]

Richard Moeckel. A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 609-620. doi: 10.3934/dcdsb.2008.10.609
[9]

Nai-Chia Chen. Symmetric periodic orbits in three sub-problems of the $N$-body problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1523-1548. doi: 10.3934/dcdsb.2014.19.1523
[10]

Marcel Guardia, Tere M. Seara, Pau Martín, Lara Sabbagh. Oscillatory orbits in the restricted elliptic planar three body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 229-256. doi: 10.3934/dcds.2017009
[11]

Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299
[12]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229
[13]

Richard Moeckel. A proof of Saari's conjecture for the three-body problem in $\R^d$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 631-646. doi: 10.3934/dcdss.2008.1.631
[14]

Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara. Determination of motion from orbit in the three-body problem. Conference Publications, 2011, 2011 (Special) : 1158-1166. doi: 10.3934/proc.2011.2011.1158
[15]

Daniel Offin, Hildeberto Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 379-392. doi: 10.3934/dcdss.2009.2.379
[16]

Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85
[17]

Florian Rupp, Jürgen Scheurle. Classification of a class of relative equilibria in three body coulomb systems. Conference Publications, 2011, 2011 (Special) : 1254-1262. doi: 10.3934/proc.2011.2011.1254
[18]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
[19]

D. Novikov and S. Yakovenko. Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems. Electronic Research Announcements, 1999, 5: 55-65.

[20]

Rehana Naz, Fazal M Mahomed, Azam Chaudhry. First integrals of Hamiltonian systems: The inverse problem. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2829-2840. doi: 10.3934/dcdss.2020121

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (84)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences