Dynamics of Kepler problem with linear drag

Abstract

We study the dynamics of Kepler problem with linear drag. We prove that motions with nonzero angular momentum have no collisions and travel from infinity to the singularity. In the process, the energy takes all real values and the angular velocity becomes unbounded. We also prove that there are two types of linear motions: capture–collision and ejection–collision. The behaviour of solutions at collisions is the same as in the conservative case. Proofs are obtained using the geometric theory of ordinary differential equations and two regularizations for the singularity of Kepler problem equation. The first, already considered in Diacu (Celest Mech Dyn Astron 75:1–15, 1999), is mainly used for the study of the linear motions. The second, the well known Levi-Civita transformation, allows to complete the study of the asymptotic values of the energy and to prove the existence of collision solutions with arbitrary energy.

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Notes

  1. 1.

    Besides the classical works by Levi-Civita it is interesting to mention Goursat’s paper (1889). The authors thank Dr. Lei Zhao for calling their attention to this paper.

  2. 2.

    It can be proved that \({\mathcal {M}}\) is a connected manifold of dimension four that is not compact and has the same type of homotopy of a 3-sphere \(\mathbb {S}^3\).

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Acknowledgments

Rafael Ortega was supported by project MTM2011-23652, Spain. Alessandro Margheri and Carlota Rebelo were supported by Fundação para a Ciência e Tecnologia, PEst, OE/MAT/UI0209/2011 and project PTDC/MAT/113383/2009.

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Correspondence to Alessandro Margheri.

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Margheri, A., Ortega, R. & Rebelo, C. Dynamics of Kepler problem with linear drag. Celest Mech Dyn Astr 120, 19–38 (2014). https://doi.org/10.1007/s10569-014-9553-8

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Keywords

  • Kepler problem
  • Linear drag
  • Collision
  • Levi-Civita transformation