Download Global phase portraits of the planar perpendicular problem of two

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