# American Institute of Mathematical Sciences

September  2021, 10(3): 657-671. doi: 10.3934/eect.2021009

## On some damped 2 body problems

 Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, LJLL, F-75005, Paris, France

Received  December 2020 Published  September 2021 Early access  January 2021

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.

Citation: Alain Haraux. On some damped 2 body problems. Evolution Equations and Control Theory, 2021, 10 (3) : 657-671. doi: 10.3934/eect.2021009
##### References:
 [1] N. Bohr, On the constitution of atoms and molecules, Philosophical Magazine, 26 (1913), 1-24. [2] H. Chabot, Georges-Louis LeSage (1724–1803): A theoretician of gravitation in search of legitimacy, Arch. Internat. Hist. Sci., 53 (2003), 157-183. [3] M. R. Edwards, Pushing gravity: New perspectives on LeSage's theory of gravitation, Revue d'Histoire des Sciences, 58 (2005), 519-520. [4] M. R. Edwards, Photon-graviton recycling as cause of gravitation, Apeiron, 14 (2007), 214-230. [5] G. Galilei, Two New Sciences, The University of Wisconsin Press, Madison, Wis., 1974. [6] A. Haraux, About Dark Matter and Gravitation, preprint, (2020), 2020070198. doi: 10.20944/preprints202007.0198.v1. [7] A. Haraux, On Carboniferous Gigantism and Atomic Shrinking, preprint, (2020), 2020110544. doi: 10.20944/preprints202011.0544.v2. [8] J. F. Harrison, A. Kaiser and J. M. VandenBrooks, Atmospheric oxygen level and the evolution of insect body size, Proceedings of the Royal Society B, 277 (2010), 1937-1946.  doi: 10.1098/rspb.2010.0001. [9] E. Hubble and M. L. Humason, The velocity-distance relation among extra-galactic nebulae, Astrophysical Journal, vol. 74, 43–80. doi: 10.1086/143323. [10] L. D. Landau and E. M. Lifschitz, Mechanics, Course of Theoretical Physics, Vol. 1, Mir Editions, Moscow, 1966. [11] A. Margheri, R. Ortega and C. Rebelo, First integrals for the Kepler problem with linear drag, Celestial Mech. Dynam. Astronom, 127 (2017), 35-48.  doi: 10.1007/s10569-016-9715-y. [12] A. Margheri, R. Ortega and C. Rebelo, On a family of Kepler problems with linear dissipation, Rend. Istit. Mat. Univ. Trieste, 49 (2017), 265-286.  doi: 10.13137/2464-8728/16216. [13] R. Parks, An Overview of Hypotheses and Supporting Evidence Regarding Drivers of Insect Gigantism in the Permo-Carboniferous, Western Washington University Reports, (2020), 1–13. [14] R. Penrose, The big bang and its dark-matter content: whence, whither, and wherefore, Found Phys., 48 (2018), 1177-1190.  doi: 10.1007/s10701-018-0162-3. [15] E. Rutherford, The scattering of $\alpha$ and $\beta$ particles by matter and the structure of the atom, E. Rutherford, F.R.S. Philosophical Magazine, 21 (1911), 669-688. [16] E. Schrödinger, An undulatory theory of the mechanics of atoms and molecules, Phys. Rev., 28 (1926), 1049-1070.  doi: 10.1103/PhysRev.28.1049. [17] F. Zwicky, The redshift of extragalactic nebulae, Helvetica Physica Acta, 6 (1933), 110-127.

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##### References:
 [1] N. Bohr, On the constitution of atoms and molecules, Philosophical Magazine, 26 (1913), 1-24. [2] H. Chabot, Georges-Louis LeSage (1724–1803): A theoretician of gravitation in search of legitimacy, Arch. Internat. Hist. Sci., 53 (2003), 157-183. [3] M. R. Edwards, Pushing gravity: New perspectives on LeSage's theory of gravitation, Revue d'Histoire des Sciences, 58 (2005), 519-520. [4] M. R. Edwards, Photon-graviton recycling as cause of gravitation, Apeiron, 14 (2007), 214-230. [5] G. Galilei, Two New Sciences, The University of Wisconsin Press, Madison, Wis., 1974. [6] A. Haraux, About Dark Matter and Gravitation, preprint, (2020), 2020070198. doi: 10.20944/preprints202007.0198.v1. [7] A. Haraux, On Carboniferous Gigantism and Atomic Shrinking, preprint, (2020), 2020110544. doi: 10.20944/preprints202011.0544.v2. [8] J. F. Harrison, A. Kaiser and J. M. VandenBrooks, Atmospheric oxygen level and the evolution of insect body size, Proceedings of the Royal Society B, 277 (2010), 1937-1946.  doi: 10.1098/rspb.2010.0001. [9] E. Hubble and M. L. Humason, The velocity-distance relation among extra-galactic nebulae, Astrophysical Journal, vol. 74, 43–80. doi: 10.1086/143323. [10] L. D. Landau and E. M. Lifschitz, Mechanics, Course of Theoretical Physics, Vol. 1, Mir Editions, Moscow, 1966. [11] A. Margheri, R. Ortega and C. Rebelo, First integrals for the Kepler problem with linear drag, Celestial Mech. Dynam. Astronom, 127 (2017), 35-48.  doi: 10.1007/s10569-016-9715-y. [12] A. Margheri, R. Ortega and C. Rebelo, On a family of Kepler problems with linear dissipation, Rend. Istit. Mat. Univ. Trieste, 49 (2017), 265-286.  doi: 10.13137/2464-8728/16216. [13] R. Parks, An Overview of Hypotheses and Supporting Evidence Regarding Drivers of Insect Gigantism in the Permo-Carboniferous, Western Washington University Reports, (2020), 1–13. [14] R. Penrose, The big bang and its dark-matter content: whence, whither, and wherefore, Found Phys., 48 (2018), 1177-1190.  doi: 10.1007/s10701-018-0162-3. [15] E. Rutherford, The scattering of $\alpha$ and $\beta$ particles by matter and the structure of the atom, E. Rutherford, F.R.S. Philosophical Magazine, 21 (1911), 669-688. [16] E. Schrödinger, An undulatory theory of the mechanics of atoms and molecules, Phys. Rev., 28 (1926), 1049-1070.  doi: 10.1103/PhysRev.28.1049. [17] F. Zwicky, The redshift of extragalactic nebulae, Helvetica Physica Acta, 6 (1933), 110-127.
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