First integrals for the Kepler problem with linear drag

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• 作者：Alessandro Margheri ; Rafael Ortega…
• 关键词：Kepler problem ; Linear drag ; First integral ; Conformally symplectic ; Global dynamics ; Runge ; Lenz ; type integral
• 刊名：Celestial Mechanics and Dynamical Astronomy
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：127
• 期：1
• 页码：35-48
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Astrophysics and Astroparticles; Dynamical Systems and Ergodic Theory; Aerospace Technology and Astronautics; Geophysics/Geodesy; Classical Mechanics;
• 出版者：Springer Netherlands
• ISSN：1572-9478
• 卷排序：127

文摘

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