# Instabilities in the Sun–Jupiter–Asteroid three body problem

## Abstract

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.

## Keywords

Hamiltonian systems Restricted problems Aubry-Mather theory Mars crossing orbits## Notes

### Acknowledgments

The authors would like to acknowledge the guidance and direction given by Vadim Kaloshin. Without his input this paper certainly would not have been possible. The authors would also like to thank Anatoly Neishtadt for his remarks.

## References

- 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
- 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
- Bernard, P.: The dynamics of pseudographs in convex Hamiltonian systems. J. Am. Math. Soc.
**21**(3), 615–669 (2008)MathSciNetMATHCrossRefGoogle Scholar - Bourgain, J., Kaloshin, V.: On diffusion in high-dimensional Hamiltonian systems. J. Funct. Anal.
**229**(1), 1–61 (2005)MathSciNetMATHCrossRefGoogle Scholar - Broucke, R., Petrovsky, T.: Area-preserving mappings and deterministic chaos for nearly parabolic motions. Celest. Mech.
**42**(1–4), 53–79 (1987)ADSGoogle Scholar - Curtis, H.: Orbital Mechanics for Engineering Students, 2nd edn. Butterworth-Heinnemann, Amsterdam (2010)Google Scholar
- Celletti, A., Chierchia, L.: KAM stability and celestial mechanics. Mem. Am. Math. Soc.
**187**(878), viii+134 (2007)Google Scholar - 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 - Fejoz, J.: Quasiperiodic motions in the planar three-body problem. J. Differ. Equ.
**183**(2), 303–341 (2002)MathSciNetMATHCrossRefGoogle Scholar - 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
- Ferraz-Mello, S.: Slow and fast diffusion in asteroid-belt resonances: a review. Celest. Mech. Dyn. Astron.
**73**, 25 (1999)MathSciNetADSMATHCrossRefGoogle Scholar - 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
- 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 - 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
- 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
- Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd edn. Addison Wesley, San Francisco (2001)Google Scholar
- Kaloshin, V.: Geometric Proof of Mather’s Connecting Theorem. Preprint. Available Online. http://www.its.caltech.edu/kaloshin/research/mather.pdf
- 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 - MacKay, R., Percival, I.: Converse KAM. Comm. Math. Phys.
**98**(4), 469–512 (1985)MathSciNetADSMATHCrossRefGoogle Scholar - Mather, J.: Variational construction of orbits of twist diffeomorphisms. J. Am. Math. Soc.
**4**(2), 207–263 (1991)MathSciNetMATHCrossRefGoogle Scholar - 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 - 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
- Moser, J.: Recent development in the theory of Hamiltonian systems. SIAM Rev.
**28**(4), 459–485 (1986)MathSciNetMATHCrossRefGoogle Scholar - 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
- 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
- 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
- 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
- Wisdom, J.: The origin of the Kirkwood gaps. Astron. J.
**87**, 577–593 (1982)MathSciNetADSCrossRefGoogle Scholar - Wisdom, J.: Chaotic behavior and the origin of the 3/1 Kirkwood gap. Icarus
**56**, 51–74 (1983)ADSCrossRefGoogle Scholar - Wisdom, J.: A pertubative treatment of motion near the 3/1 commensurability. Icarus
**63**, 272–289 (1985)ADSCrossRefGoogle Scholar - Wilczak, D., Zgliczynski, P.: The \(C^r\) Lohner-algorithm. arXiv:0704.0720v1 (2007)Google Scholar
- Xia, J.: Arnold Diffusion and Instabilities in Hamiltonian Systems. Preprint. Available Online. http://www.math.northwestern.edu/xia/preprint/arndiff.ps