The Three-Body Problem

The most celebrated problem in celestial mechanics is the so-called three-body problem. First elucidated by Lagrange, this problem focuses on the determination of the allowed class of periodic motions for a massless particle orbiting a binary system. In this case, the motion is determined by the gravitational and centrifugal accelerations and also the Coriolis force. A closed form analytic solution is possible in only one case, that of equal masses in a circular orbit. This so-caUed restricted three-body problem can be specified by the curves of constant potential, also called the zero velocity surfaces. Consider a binary with a coplanar orbit for the third mass. In this case, a local coordinate system (C, r]) is defined as centered at (a, 1 — a) so that the equations of motion are 

Here the gravitational potential, 1 , is the Roche potential already discussed. The assumption required for this potential is that the two massive bodies are in a circular orbit about the center of mass. In the absence of eccentricity, stable orbits are possible in several regions of the orbital plane. These are defined by the condition that V 4 = 0 and are critical points in the solution of the equations of motion. These are stationary in the rotating frame. In the presence of eccentricity, they oscillate and produce a loss of stability, as we shall explain in Section IV.C. 

Several critical points in the three-body potential dominate the motion of particles. They are specified as the points at which the gravitational acceleration vanishes. 

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Once electron repulsion is taken into account, this separation of a many-electron wavefunction into a product of one-electron wavefunctions (orbitals) is no longer possible. This is not a failing of quanmm mechanics scientists and engineers reach similar conclusions whenever they have to deal with problems involving three or more mutually interacting particles. We speak of the three-body problem. 

Thus, astronomers also suffer from the three-body problem when they try to study the motion of the planets round the sun. They are lucky in that the gravitational force between bodies A and B goes as... 

The transition from a macroscopic description to the microscopic level is always a complicated mathematical problem (the so-called many-particle problem) having no universal solution. To illustrate this point, we recommend to consider first the motion of a single particle and then the interaction of two particles, etc. The problem is well summarized in the following remark from a book by Mattuck [18] given here in a shortened form. For the Newtonian mechanics of the 18th century the three-body problem was unsolvable. The general theory of relativity and quantum electrodynamics created unsolvable two-body and single-body problems. Finally, for the modem quantum field... 

E. Nielsen, D.V. Fedorov, A.S. Jensen, E. Garrido, The three-body problem with short-range interactions, Phys. Rep. 347 (2001) 373. 

The two-body dynamics described in the preceding section has been useful in introducing a number of important concepts, and we have obtained valuable insights concerning the angular distribution of scattered particles. However, there is obviously no way to faithfully describe a chemical reaction in terms of only two interacting particles at least three particles are required. Unfortunately, the three-body problem is one for which no analytic solution is known. Accordingly, we must use numerical analysis and computers to solve this problem.7... 

To determine the heat flows in a problem of this type, the values of the radi-osities must be calculated. This may be accomplished by performing standard methods of analysis used in dc circuit theory. The most convenient method is an application of Kirchhoffs current law to the circuit, which states that the sum of the currents entering a node is zero. Example 8-5 illustrates the use of the method for the three-body problem. 

None of these two-body methods has yet been extended to the many-body problem. A contribution by Kynch (1959) applied a reflection scheme to the three-body problem, although explicit details were not provided. 

A problem of great practical importance is the three-body problem with non-Keplerian forces and a repulsive core. The triatomic problem has widespread applications in reactive scattering and triatomic isomerization. There are many studies dealing with experiments and theory. The principal problem for reactive scattering has always been the calculation of a reliable potential function on which the triatomic dynamics takes place. However, quantum dynamics on those... 

The three-body problem appears in various physical and chemical systems—that is, celestial systems (e.g., the sun, the earth, and the moon), atomic systems (e.g., two electrons and one nucleus), and molecular systems (e.g., D + H2 DH + H reaction). Due to historical reasons, the three-body problem in celestial mechanics is the oldest. In order for our ancestors to make the calender, they observed the motion of the sun and the moon for agriculmral and fishery purposes and also for daily life. After Copernicus, they knew that the earth itself moves. But they did not know the law of the motion of stars and planets. By... 

The stability of the solar system is one of the most important unsettled questions of classical mechanics. Even a simplified version of the solar system, the three-body problem, presents a formidable challenge. An important breakthrough occurred when Poincare, with some assistance from his Swedish colleague Pragmen, proved in 1892 that, apart from some notable exceptions, the three-body problem does not possess a complete set of integrals of the motion. Thus, in modern parlance, the three-body problem is chaotic. 

Scattering from alkali-metal atoms is understood as the three-body problem of two electrons interacting with an inert core. The electron—core potentials are frozen-core Hartree—Fock potentials with core polarisation being represented by a further potential (5.82). 

This problem is called the three-body problem by the people who study such things (the theoreticians of quantum mechanics). When you have two particles in motion that attract each other, you can describe the situation with an equation. But when you have three particles, and there are attractions and repulsions, and all these particles are in motion, there are too many things going on for one neat equation. The problem is one of clouds a cloud exists and we can point to it and measure it, but to predict in advance just where it will be and what form it will take is not possible. There are too many factors, too many variables, many of which are unknown or unknowable. This problem is at the heart of the probability approach to atomic structure. 

Difficulties arise, however, when one considers more than two bodies. The motions in a system of three interacting bodies, defining (sensibly enough) the three-body problem, cannot fully be treated analytically, meaning that one cannot derive on paper a single equation to predict the positions and velocities of the three bodies in the system at some arbitrary time in the future. An analytic solution is likewise impossible for a larger system of some number N of objects, and this seemingly intractable situation is called the N-body problem. 

In this paper we consider some coordinate sets used for the treatment in classical and quantum mechanics of the motion of three particles in space. The alternative sets of coordinate systems for the three-body problem have been studied extensively [1-5] and a good choice of the coordinate systems is of crucial importance. Key references for the basic theory are 11 1,6], where also history is sketched and credits are given. 

In the three-body problem we can write down two Jacobi vectors, one (xq,) is the interparticle distance between two particles and the other (Xa) connects their center-of-mass to the third particle. So, the choice of the Jacobi vectors is not unique [1]. Here we will consider Xq, as the vector from the particle B to the particle C, and X as the vector from the particle A to the center-of-mass of the BC couple (see Figure 2). 

The exact value cannot be calculated because of the insolubility of the three-body problem of two electrons and a doubly charged nucleus. Nevertheless, we do know the exact energy necessary to remove the second electron... 

We will use the classical Jacobi decomposition of the three-body problem (Figure 1), with ... 

Easton R. (1971). Some topology of the three-body problem. Journal of differential equations 10, p. 371-377. 

Marchal C. (1990). The three-body problem,. Elsevier Science Publisher B.V. 

We begin by summarizing the equations of motion of the three-body problem with two small masses in the form of two weakly coupled Kepler motions, valid in two or three dimensions. 

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