# First integrals for the Kepler problem with linear drag

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## Abstract

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 \(\frac{1}{n}\).

## Keywords

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

### Acknowledgments

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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