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Periodic Solutions of an Indefinite Singular Equation Arising from the Kepler Problemon the Sphere

Published online by Cambridge University Press:  20 November 2018

Robert Hakl  and
Manuel Zamora
Robert Hakl
Affiliation:
Institute of Mathematics, AS CR, Czech Republic e-mail: [email protected]
Manuel Zamora
Affiliation:
Departamento de Matemática, Grupo de Investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile e-mail: [email protected]
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Abstract

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We study a second-order ordinary differential equation coming from the Kepler problem on ${{\mathbb{S}}^{2}}$. The forcing term under consideration is a piecewise constant with singular nonlinearity that changes sign. We establish necessary and sufficient conditions to the existence and multiplicity of $T$-periodic solutions.

Keywords

34B1634C2570F0570F15singular differential equationindefinite singularityperiodic solutionKepler problem on 𝕊1degree theory
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 1 , 01 February 2018 , pp. 173 - 190
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Bolyai, W. and Bolyai, J., Geometrische Untersuchungen. Hrsg. P. Stäckel, Teubner, Leipzig, 1913.Google Scholar
[2] Bravo, J. L. and Torres, P. J., Periodic solutions of a singular equation with indefinite weight. Adv. Nonlinear Stud. 10(2010), no. 4, 927–938.http://dx.doi.org/10.1515/ans-2010-0410 Google Scholar
[3] Fonda, A., Manàsevich, R., and Zanolin, F., Subharmonic solutions for some second-order differential equations with singularities. SIAM J. Math. Anal. 24(1993), no. 5,1294–1311. http://dx.doi.Org/10.1137/0524074 Google Scholar
[4] Franco-Pérez, L. and Pérez-Chavela, E., Global symplectic regularization for some restricted 3-body problems on . Nonlinear Anal. 71(2009), no. 11, 5131–5143.http://dx.doi.Org/10.1016/j.na.2009.03.089 Google Scholar
[5] Kozlov, V. V. and Harin, A. O., Kepler's problem in constant curvature spaces. Celestial Mech. Dynam. Astronom. 54(1992), no. 4, 393–399. http://dx.doi.Org/10.1007/BF00049149 Google Scholar
[6] Lobachevsky, N., The new foundations of geometry with full theory of parallels. (Russian) In: Collected Works, vol. 2, p. 159. GITTL, 1835–1838.Google Scholar
[7] Nakajima, F. and Seifert, G., The number of periodic solutions of 2-dimensional periodic systems. J. Differential Equations 49(1983), 430–440. http://dx.doi.Org/10.1016/0022-0396(83)90005-0 Google Scholar
[8] Shchepetilov, A. V., Nonintegrability of the two-body problem in constant curvature spaces. J. Phys. A 39(2006), 5787–5806.http://dx.doi.Org/10.1088/0305-4470/39/20/011 Google Scholar
[9] Serret, P. J., Théorie nouvelle géométrique et mécanique des lignes a double courbure. Librave de Mallet-Bachelier, Paris, 1860.Google Scholar