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Astrophysics and Space Science

, 363:255 | Cite as

Global dynamics of the Buckingham’s two-body problem

  • Jaume Llibre
  • Claudia Valls
  • Claudio VidalEmail author
Original Article
  • 51 Downloads

Abstract

The equations of motion of the Buckingham system are the ones of a two-body problem defined by the Hamiltonian
$$ H= \frac{1}{2} \bigl(p_{x}^{2}+p_{y}^{2} \bigr) + A e^{-B \sqrt{x ^{2}+y^{2}}}-\frac{M}{ (x^{2}+y^{2} )^{3}}, $$
where \(A\), \(B\) and \(M\) are positive constants. The angular momentum \(p_{\theta }:= x p_{y}- y p_{x}\) and this Hamiltonian are two independent first integrals in involution. We denote by \(I_{h}\) (respectively, \(I_{c}\)) the set of points of the phase space where \(H\) (respectively, \(p_{\theta }\)) takes the value \(h\) (respectively, \(c\)). Due to the fact that \(H\) and \(p_{\theta }\) are first integrals, the sets \(I_{h}\) and \(I_{hc}= I_{h} \cap I_{c}\) are invariant under the flow of the Buckingham systems. We describe the global dynamics of the Buckingham system describing the foliation of its phase space by the invariant sets \(I_{h}\), the foliation of the invariant set \(I_{h}\) by its invariant subsets \(I_{hc}\), and the foliation of invariant set \(I_{hc}\) by the orbits of the system.

Keywords

Buckingham equations Hill regions Global dynamics 

Notes

Acknowledgements

This work is supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación within the grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics-H2020-MSCA-RISE-2017-777911. The second author is partially supported by FCT/Portugal through the project UID/MAT/04459/2013. The third author is partially supported by Fondecyt project 1180288.

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain
  2. 2.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisboaPortugal
  3. 3.Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de CienciasUniversidad del Bío-BíoConcepciónChile

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