• Previous Article
    Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces
  • EECT Home
  • This Issue
  • Next Article
    Results on controllability of non-densely characterized neutral fractional delay differential system
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  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 & Control Theory, doi: 10.3934/eect.2021009
References:
[1]

N. Bohr, On the constitution of atoms and molecules, Philosophical Magazine, 26 (1913), 1-24.   Google Scholar

[2]

H. Chabot, Georges-Louis LeSage (1724–1803): A theoretician of gravitation in search of legitimacy, Arch. Internat. Hist. Sci., 53 (2003), 157-183.   Google Scholar

[3]

M. R. Edwards, Pushing gravity: New perspectives on LeSage's theory of gravitation, Revue d'Histoire des Sciences, 58 (2005), 519-520.   Google Scholar

[4]

M. R. Edwards, Photon-graviton recycling as cause of gravitation, Apeiron, 14 (2007), 214-230.   Google Scholar

[5]

G. Galilei, Two New Sciences, The University of Wisconsin Press, Madison, Wis., 1974.  Google Scholar

[6]

A. Haraux, About Dark Matter and Gravitation, preprint, (2020), 2020070198. doi: 10.20944/preprints202007.0198.v1.  Google Scholar

[7]

A. Haraux, On Carboniferous Gigantism and Atomic Shrinking, preprint, (2020), 2020110544. doi: 10.20944/preprints202011.0544.v2.  Google Scholar

[8]

J. F. HarrisonA. 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.  Google Scholar

[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.  Google Scholar

[10]

L. D. Landau and E. M. Lifschitz, Mechanics, Course of Theoretical Physics, Vol. 1, Mir Editions, Moscow, 1966. Google Scholar

[11]

A. MargheriR. 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.  Google Scholar

[12]

A. MargheriR. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

F. Zwicky, The redshift of extragalactic nebulae, Helvetica Physica Acta, 6 (1933), 110-127.   Google Scholar

show all references

References:
[1]

N. Bohr, On the constitution of atoms and molecules, Philosophical Magazine, 26 (1913), 1-24.   Google Scholar

[2]

H. Chabot, Georges-Louis LeSage (1724–1803): A theoretician of gravitation in search of legitimacy, Arch. Internat. Hist. Sci., 53 (2003), 157-183.   Google Scholar

[3]

M. R. Edwards, Pushing gravity: New perspectives on LeSage's theory of gravitation, Revue d'Histoire des Sciences, 58 (2005), 519-520.   Google Scholar

[4]

M. R. Edwards, Photon-graviton recycling as cause of gravitation, Apeiron, 14 (2007), 214-230.   Google Scholar

[5]

G. Galilei, Two New Sciences, The University of Wisconsin Press, Madison, Wis., 1974.  Google Scholar

[6]

A. Haraux, About Dark Matter and Gravitation, preprint, (2020), 2020070198. doi: 10.20944/preprints202007.0198.v1.  Google Scholar

[7]

A. Haraux, On Carboniferous Gigantism and Atomic Shrinking, preprint, (2020), 2020110544. doi: 10.20944/preprints202011.0544.v2.  Google Scholar

[8]

J. F. HarrisonA. 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.  Google Scholar

[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.  Google Scholar

[10]

L. D. Landau and E. M. Lifschitz, Mechanics, Course of Theoretical Physics, Vol. 1, Mir Editions, Moscow, 1966. Google Scholar

[11]

A. MargheriR. 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.  Google Scholar

[12]

A. MargheriR. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

F. Zwicky, The redshift of extragalactic nebulae, Helvetica Physica Acta, 6 (1933), 110-127.   Google Scholar

[1]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[2]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[3]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[4]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[5]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[6]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[7]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[8]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[9]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475

[10]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[11]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[12]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[13]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[14]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

[15]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[16]

Palash Sarkar, Subhadip Singha. Verifying solutions to LWE with implications for concrete security. Advances in Mathematics of Communications, 2021, 15 (2) : 257-266. doi: 10.3934/amc.2020057

[17]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020210

[18]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[19]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[20]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

2019 Impact Factor: 0.953

Article outline

[Back to Top]