Celestial Mechanics and Dynamical Astronomy

, Volume 127, Issue 1, pp 35–48 | Cite as

First integrals for the Kepler problem with linear drag

  • Alessandro MargheriEmail author
  • Rafael Ortega
  • Carlota Rebelo
Original Article


In this work we consider the Kepler problem with linear drag, and prove the existence of a continuous vector-valued first integral, obtained taking the limit as \(t\rightarrow +\infty \) of the Runge–Lenz vector. The norm of this first integral can be interpreted as an asymptotic eccentricity \(e_{\infty }\) with \(0\le e_{\infty } \le 1\). The orbits satisfying \(e_{\infty } <1\) approach the singularity by an elliptic spiral and the corresponding solutions \(x(t)=r(t)e^{i\theta (t)}\) have a norm r(t) that goes to zero like a negative exponential and an argument \(\theta (t)\) that goes to infinity like a positive exponential. In particular, the difference between consecutive times of passage through the pericenter, say \(T_{n+1} -T_n\), goes to zero as \(\frac{1}{n}\).


Kepler problem Linear drag First integral Conformally symplectic Global dynamics Runge-Lenz-type integral 



Alessandro Margheri: Supported by Fundação para a Ciência e Tecnologia, UID/MAT/04561/2013. Rafael Ortega: Supported by project MTM2014–52232–P, Spain. Carlota Rebelo: Supported by Fundação para a Ciência e Tecnologia, UID/MAT/04561/2013.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Alessandro Margheri
    • 1
    Email author
  • Rafael Ortega
    • 2
  • Carlota Rebelo
    • 1
  1. 1.Fac. Ciências da Univ. de Lisboa e, Centro de MatemáticaAplicações Fundamentais e Investigação OperacionalLisbonPortugal
  2. 2.Departamento de Matemática AplicadaUniversidad de GranadaGranadaSpain

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