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JOURNAL OF MATHEMATICAL PHYSICS 50, 042903 共2009兲 Global phase portraits of the planar perpendicular problem of two fixed centers Lidia Jiménez–Lara,1,a兲 Jaume Llibre,2,b兲 and Martín Vargas1 1 Departamento de Física, Universidad Autónoma Metropolitana–Iztapalapa, P.O. Box 55-534, Mexico, D.F. 09340, Mexico 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain 共Received 20 October 2008; accepted 15 February 2009; published online 28 April 2009兲 We study the global phase portrait of the classical problem of an electron in the electrostatic field of two protons that we assume fixed to symmetric distances on the x3 axis. The general problem can be formulated as an integrable Hamiltonian system of three degrees of freedom, but we restrict our study to the invariant planar case that is equidistant to the two fixed centers. This is a two degrees of freedom problem with two constants of motion, the energy and the angular momentum, denoted by H and C, respectively, which are independent and in involution. We describe the foliation of the four-dimensional phase space by the invariant sets of constant energy Ih and we characterize their topology. We also describe the foliation of each energy level Ih by the invariant sets Ihc, and we classify the topology of Ihc and the flow on these invariant sets. In this way we provide a global qualitative description of the motion. We also compare our results with the existing published results. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3097195兴 I. INTRODUCTION We study a particular three-body problem consisting of one very light particle at the position 共q1 , q2 , q3兲 that does not influence the motion of the other two, which are considered fixed on the q3 axis of an inertial frame and located to symmetric distances from the origin. This problem is known as the two fixed centers or two centers of force and was studied for the first time by Euler.8 As we will assume that the two fixed particles are equal, it is better to think that the problem involves electrical forces predominating on the gravitational ones, as in the case of protons and electrons in a molecule. This model can be used to describe classically the motion of an electron in the field of two protons, as the hydrogen molecule ion. The quantum solution is rated by Gutzwiller9 as one of the most important in quantum mechanics because it can explain the chemical bond between two protons by a single electron. A refined treatment using the more general phase integral method can be found in the papers in Refs. 3–5. We consider the two protons fixed at the q3 axis at the points A = 共0 , 0 , −a兲 and B = 共0 , 0 , a兲 in a Cartesian coordinate system. An electron at 共q1 , q2 , q3兲 position interacts with both protons by mean of the Coulomb law. The Hamiltonian for the electron is H̄ = 1 2 ke2 ke2 共p1 + p22 + p23兲 − 2 − 2 2 2 冑q1 + q2 + 共q3 − a兲2 冑q1 + q2 + 共q3 + a兲2 , 2m where m, e, qi, and pi are the mass, charge, i position, and i linear momentum of the electron and k is the electrostatic constant. We introduce the variables a兲 Electronic mail: [email protected]. Electronic mail: [email protected]. b兲 0022-2488/2009/50共4兲/042903/10/$25.00 50, 042903-1 © 2009 American Institute of Physics Downloaded 29 May 2009 to 150.214.204.48. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp