Preprint Brief Report Version 3 Preserved in Portico This version is not peer-reviewed

# On Some Damped 2 Body Problems

Version 1 : Received: 19 December 2020 / Approved: 21 December 2020 / Online: 21 December 2020 (11:19:42 CET)
Version 2 : Received: 23 December 2020 / Approved: 23 December 2020 / Online: 23 December 2020 (17:25:05 CET)
Version 3 : Received: 4 January 2021 / Approved: 5 January 2021 / Online: 5 January 2021 (10:07:44 CET)

How to cite: Haraux, A. On Some Damped 2 Body Problems. Preprints 2020, 2020120510 (doi: 10.20944/preprints202012.0510.v3). Haraux, A. On Some Damped 2 Body Problems. Preprints 2020, 2020120510 (doi: 10.20944/preprints202012.0510.v3).

## Abstract

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 which were investigated thoroughly by R. Ortega and his co-authors between 2014 and 2017, in particular all solutions are bounded and tend to $0$ for $t$ large, some of them with asymptotically spiraling exponentially fast convergence to the center. We provide explicit estimates for the bounds in the general case that we refine under specific restrictions on the initial state, and we give a formal calculation which could be used to determine practically some special asymptotically spiraling orbits. Besides, a related model with exponentially damped central charge or mass gives some explicit exponentially decaying solutions which might help future investigations. An atomic contraction hypothesis related to the asymptotic dying off of solutions proven for the dissipative model might give a solution to some intriguing phenomena observed in paleontology, familiar electrical devices and high scale cosmology

## Subject Areas

gravitation; singular potential; global solutions; spiraling orbit

Comment 1
Commenter: Alain Haraux
Commenter's Conflict of Interests: Author
Comment: Changes in Section 3-4, and accordingly in the abstract, introduction and references to take account of recent important results by R. Ortega and his co-authors.
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