Preprint Brief Report Version 2 Preserved in Portico This version is not peer-reviewed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.v2). Haraux, A. On Some Damped 2 Body Problems. Preprints 2020, 2020120510 (doi: 10.20944/preprints202012.0510.v2).

## 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. 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. 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: I corrected some formulas in subsection 4.1 and gave them numbers. I also adapted the corresponding text.
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