Celestial Mechanics and Dynamical Astronomy

, Volume 115, Issue 3, pp 233–259 | Cite as

Instabilities in the Sun–Jupiter–Asteroid three body problem

Original Article


We consider dynamics of a Sun–Jupiter–Asteroid system, and, under some simplifying assumptions, show the existence of instabilities in the motions of an asteroid. In particular, we show that an asteroid whose initial orbit is far from the orbit of Mars can be gradually perturbed into one that crosses Mars’ orbit. Properly formulated, the motion of the asteroid can be described as a Hamiltonian system with two degrees of freedom, with the dynamics restricted to a “large” open region of the phase space reduced to an exact area preserving map. Instabilities arise in regions where the map has no invariant curves. The method of MacKay and Percival is used to explicitly rule out the existence of these curves, and results of Mather abstractly guarantee the existence of diffusing orbits. We emphasize that finding such diffusing orbits numerically is quite difficult, and is outside the scope of this paper.


Hamiltonian systems Restricted problems Aubry-Mather theory Mars crossing orbits 


  1. Arnol’d, V., Kozlov, V., Neishtadt, A.: I.: Mathematical aspects of classical and celestial mechanics. Dynamical systems. III. Translated from the Russian original by E. Khukhro. Third edition. Encyclopaedia of Mathematical Sciences, 3. Springer, Berlin (2006)Google Scholar
  2. Bangert, V.: Mather sets for twist maps and geodesics on tori. Dynamics reported, vol. 1, 1–56, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester (1988)Google Scholar
  3. Bernard, P.: The dynamics of pseudographs in convex Hamiltonian systems. J. Am. Math. Soc. 21(3), 615–669 (2008)MathSciNetMATHCrossRefGoogle Scholar
  4. Bourgain, J., Kaloshin, V.: On diffusion in high-dimensional Hamiltonian systems. J. Funct. Anal. 229(1), 1–61 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. Broucke, R., Petrovsky, T.: Area-preserving mappings and deterministic chaos for nearly parabolic motions. Celest. Mech. 42(1–4), 53–79 (1987)ADSGoogle Scholar
  6. Curtis, H.: Orbital Mechanics for Engineering Students, 2nd edn. Butterworth-Heinnemann, Amsterdam (2010)Google Scholar
  7. Celletti, A., Chierchia, L.: KAM stability and celestial mechanics. Mem. Am. Math. Soc. 187(878), viii+134 (2007)Google Scholar
  8. Chenciner, A., Llibre, J.: A note on the existence of invariant punctured tori in the planar circular restricted three-body problem. Ergod. Theor. Dyn. 8, 63–72 (1988)MathSciNetCrossRefGoogle Scholar
  9. Fejoz, J.: Quasiperiodic motions in the planar three-body problem. J. Differ. Equ. 183(2), 303–341 (2002)MathSciNetMATHCrossRefGoogle Scholar
  10. Fejoz, J., Guardia, M., Kaloshin, V., Roldan, P.: Diffusion along mean motion resonance in the restricted planar three-body problem. arXiv:1109.2892v1 (2011)Google Scholar
  11. Ferraz-Mello, S.: Slow and fast diffusion in asteroid-belt resonances: a review. Celest. Mech. Dyn. Astron. 73, 25 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  12. Gole, C..: Symplectic twist maps. Global variational techniques. Advanced series in nonlinear dynamics, 18. World Scientific Publishing Co., Inc., River Edge, xviii+305 pp. (2001) ISBN: 981-02-0589-9Google Scholar
  13. Galante, J., Kaloshin, V.: Destruction of invariant curves in the restricted circular planar three body problem using comparison of action. Duke Math. J. 159(2), 275–327 (2011)MathSciNetMATHCrossRefGoogle Scholar
  14. Galante, J., Kaloshin, V.: Construction of a twisting coordinate system for the restricted circular planar three body problem. Manuscript. Available at http://www.terpconnect.umd.edu/vkaloshi/papers/Twist-spreading-Joseph.pdf
  15. Galante, J., Kaloshin, V.: Destruction of invariant curves in the restricted circular planar three body problem using ordering condition. Manuscript. Avaliable at http://www.terpconnect.umd.edu/vkaloshi/papers/localization-joseph.pdf
  16. Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd edn. Addison Wesley, San Francisco (2001)Google Scholar
  17. Kaloshin, V.: Geometric Proof of Mather’s Connecting Theorem. Preprint. Available Online. http://www.its.caltech.edu/kaloshin/research/mather.pdf
  18. Liao, X., Saari, D.G.: Instability and diffusion in the elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 70(1), 23–39 (1998)MathSciNetADSMATHCrossRefGoogle Scholar
  19. MacKay, R., Percival, I.: Converse KAM. Comm. Math. Phys. 98(4), 469–512 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  20. Mather, J.: Variational construction of orbits of twist diffeomorphisms. J. Am. Math. Soc. 4(2), 207–263 (1991)MathSciNetMATHCrossRefGoogle Scholar
  21. Mather, J.: Differentiability of the minimal average action as a function of the rotation number. Bol. Soc. Brasil. Mat. 21, 59–70 (1990)MathSciNetMATHCrossRefGoogle Scholar
  22. Mather, J., Forni, G.: Action minimizing orbits in Hamiltonian systems. Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), 92–186, Lecture Notes in Math., 1589, Springer, Berlin (1994)Google Scholar
  23. Moser, J.: Recent development in the theory of Hamiltonian systems. SIAM Rev. 28(4), 459–485 (1986)MathSciNetMATHCrossRefGoogle Scholar
  24. Moser, J.: Stable and random motions in dynamical systems. With special emphasis on celestial mechanics. Reprint of the 1973 original. With a foreword by Philip J. Holmes. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ (2001)Google Scholar
  25. Rogel, J.V.: Early aubry-mather theory. Informal talks delivered at the Summer colloquium of the computational science department at the National University of Singapore (2001)Google Scholar
  26. Siburg, K.F.: The principle of least action in geometry and dynamics. Lecture Notes in Mathematics, Springer-Verlag, Berlin, xii+128 pp. (2004) ISBN: 3-540-21944-7Google Scholar
  27. Siegel, C., Moser, J.: Lectures on celestial mechanics. Translation by Charles I. Kalme. Die Grundlehren der mathematischen Wissenschaften, Band 187. Springer-Verlag, New York (1971)Google Scholar
  28. Wisdom, J.: The origin of the Kirkwood gaps. Astron. J. 87, 577–593 (1982)MathSciNetADSCrossRefGoogle Scholar
  29. Wisdom, J.: Chaotic behavior and the origin of the 3/1 Kirkwood gap. Icarus 56, 51–74 (1983)ADSCrossRefGoogle Scholar
  30. Wisdom, J.: A pertubative treatment of motion near the 3/1 commensurability. Icarus 63, 272–289 (1985)ADSCrossRefGoogle Scholar
  31. Wilczak, D., Zgliczynski, P.: The \(C^r\) Lohner-algorithm. arXiv:0704.0720v1 (2007)Google Scholar
  32. Xia, J.: Arnold Diffusion and Instabilities in Hamiltonian Systems. Preprint. Available Online. http://www.math.northwestern.edu/xia/preprint/arndiff.ps

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Department of MathematicsPenn State UniversityUniversity Park, State CollegeUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations