Preprint Brief Report Version 2 Preserved in Portico This version is not peer-reviewed

On a Linearly Damped 2 Body Problem

Version 1 : Received: 11 December 2020 / Approved: 14 December 2020 / Online: 14 December 2020 (09:11:53 CET)
Version 2 : Received: 15 December 2020 / Approved: 16 December 2020 / Online: 16 December 2020 (10:09:41 CET)

How to cite: Haraux, A. On a Linearly Damped 2 Body Problem. Preprints 2020, 2020120309. Haraux, A. On a Linearly Damped 2 Body Problem. Preprints 2020, 2020120309.


The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover we prove that all bounded solutions tend to $0$ for $t$ large, and some formal calculations suggest the existence of special orbits with an asymptotically spiraling exponentially fast convergence to the center.


gravitation; singular potential; global solutions; spiraling orbit


Physical Sciences, Mathematical Physics

Comments (1)

Comment 1
Received: 16 December 2020
Commenter: Alain Haraux
Commenter's Conflict of Interests: Author
Comment: Correction of a misprint+ New Section and New Theorem on convergence of bounded solutions to $0$. Te abstract is modified accordingly. 
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