Chaos in Planar, Circular, Restricted Three-Body Problem

ABSTRACT

In this article we analyze the motion of a test particle of a planar, circular, restricted three-body problem in resonance, using the Kustaanheimo-Stiefel formalism. We show that a good qualitative description of the motion can be reduced to three simple equations for semi-major axis, eccentricity and resonance angle. Studying these equations reveals the onset of chaos, and sheds a new light on its weak nature. The 7:4 resonance is used as an example.

Cite this paper

J. Vrbik, "Chaos in Planar, Circular, Restricted Three-Body Problem,"*Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 40-45. doi: 10.4236/am.2013.41008.

J. Vrbik, "Chaos in Planar, Circular, Restricted Three-Body Problem,"

References

[1] J. Henrard and N. D. Caranicolas, “Motion near the 3/1 Resonance of the Planar Elliptic Restricted Three Body Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 47, No. 2, 1989, pp. 99-121. doi:10.1007/BF00051201

[2] C. D. Murray and S. F. Dermott, “Solar System Dynamics,” Cambridge University Press, Cambridge, 1999.

[3] N. Haghighipour, “Resonance Dynamics and Partial Averaging in a Restricted Three-Body System,” Journal of Mathematical Physics, Vol. 43, No. 7, 2002, pp. 3678-3694. doi:10.1063/1.1482148

[4] J. Vrbik, “New Methods of Celestial Mechanics,” Bentham Science Publishers, Sharjah, 2010.

[5] A. Celletti, A. Chessa, J. Hadjidemetriou and G. B. Valsecchi, “A Systematic Study of the Stability of Symmetric Periodic Orbits in the Planar, Circular, Restricted Three-Body Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 83, No. 1-4, 2002, pp. 239-255. doi:10.1023/A:1020111621542

[6] J. Vrbik, “Kepler Problem with Time-Dependent and Resonant Perturbations,” Journal of Mathematical Physics, Vol. 48, No. 5, 2007, pp. 1-13. doi:10.1063/1.2729369

[1] J. Henrard and N. D. Caranicolas, “Motion near the 3/1 Resonance of the Planar Elliptic Restricted Three Body Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 47, No. 2, 1989, pp. 99-121. doi:10.1007/BF00051201

[2] C. D. Murray and S. F. Dermott, “Solar System Dynamics,” Cambridge University Press, Cambridge, 1999.

[3] N. Haghighipour, “Resonance Dynamics and Partial Averaging in a Restricted Three-Body System,” Journal of Mathematical Physics, Vol. 43, No. 7, 2002, pp. 3678-3694. doi:10.1063/1.1482148

[4] J. Vrbik, “New Methods of Celestial Mechanics,” Bentham Science Publishers, Sharjah, 2010.

[5] A. Celletti, A. Chessa, J. Hadjidemetriou and G. B. Valsecchi, “A Systematic Study of the Stability of Symmetric Periodic Orbits in the Planar, Circular, Restricted Three-Body Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 83, No. 1-4, 2002, pp. 239-255. doi:10.1023/A:1020111621542

[6] J. Vrbik, “Kepler Problem with Time-Dependent and Resonant Perturbations,” Journal of Mathematical Physics, Vol. 48, No. 5, 2007, pp. 1-13. doi:10.1063/1.2729369