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


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.


Buckingham equations Hill regions Global dynamics 



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.


  1. Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin, Reading (1978) zbMATHGoogle Scholar
  2. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin (1978) CrossRefGoogle Scholar
  3. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Dynamical Systems III. Enyclopaedia of Mathematical Sciences. Springer, Berlin (1978) Google Scholar
  4. Buckingham, R.A.: The classical equation of state of gaseous Helium, Neon and Argon. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 168, 264–283 (1938) ADSGoogle Scholar
  5. Hirsch, M.W.: Differential Topology. Graduate Texts in Math., vol. 33. Springer, New York (1976) zbMATHGoogle Scholar
  6. Lennard-Jones, J.: Cohesion. J. Proc. Phys. Soc. 43, 461–485 (1931) ADSCrossRefGoogle Scholar
  7. Llibre, J., Nunes, A.: Separatrix Surfaces and Invariant Manifolds of a Class of Integrable Hamiltonian Systems and Their Perturbations. Memoirs of the Amer. Math. Soc., vol. 513 (1994) zbMATHGoogle Scholar
  8. Meyer, K.R., Hall, G.R., Offin, D.: Introduction to Hamiltonian Dynamical Systems and the \(N\)-Body Problem, 2nd edn. Springer, Berlin (2009) zbMATHGoogle Scholar
  9. Mioc, V., Popescu, E., Popescu, N.A.: Grous of symmetries in Lennard-Jones-type-problems. Rom. Astron. J. 18, 151–166 (2008) ADSGoogle Scholar
  10. Popescu, E.: Equilibrium points in a Buckingham type problem. Rom. Astron. J. 25, 149–156 (2015) ADSGoogle Scholar
  11. Popescu, E., Pricopi, D.: Global flow in the generalized Buckhingham’s two-body problem. Astrophys. Space Sci. 362, 78–85 (2017) ADSCrossRefGoogle Scholar
  12. Pricopi, D., Popescu, E.: Phase-space structure of the Buckhingham’s two-body problem. Astrophys. Space Sci. 361, 190–196 (2016) ADSCrossRefGoogle Scholar

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© 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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