Celestial Mechanics and Dynamical Astronomy

, Volume 120, Issue 1, pp 19–38 | Cite as

Dynamics of Kepler problem with linear drag

  • Alessandro Margheri
  • Rafael Ortega
  • Carlota Rebelo
Original Article


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.


Kepler problem Linear drag Collision Levi-Civita transformation 



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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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Alessandro Margheri
    • 1
  • Rafael Ortega
    • 2
  • Carlota Rebelo
    • 1
  1. 1.Centro de Matemática e Aplicações FundamentaisFaculdade de Ciências Universidade de LisboaLisboaPortugal
  2. 2.Departamento de Matemática AplicadaUniversidad de GranadaGranadaSpain

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