Three-body problems

Smith F T 1962 A symmetric representation for three-body problems. I. Motion in a plane J. Math. Phys. 3 735-48... 

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 quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

According to modern science, all various kinds of matter consist essentially of a few types of elementary particles combined together in different ways. Since these particles do not obey the laws of classical physics but the laws of modern wave mechanics, the problem of the constitution of matter is a quantum-mechanical many-particle problem of a much higher degree of complexity than even the famous classical three-body problem. 

F. Gabern and A. Jorba, Generalizing the restricted three-body problem the bianular and tricircular coherent problems, Astron. Astrophys. 420, 751 (2004). 

The full Hamiltonian for three body problem is given as... 

The H7+ molecule-ion, which consists of two protons and one electron, represents an even simpler case of a covalent bond, in which only one electron is shared between the two nuclei. Even so, it represents a quantum mechanical three-body problem, which means that solutions of the wave equation must be obtained by iterative methods. The molecular orbitals derived from the combination of two Is atomic orbitals serve to describe the electronic configurations of the four species H2+, H2, He2+ and He2. 

More complicated behaviors are expected for triatomic molecules (i.e., for three-body problems). In general, the analysis is facilitated by the fact that... 

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... 

The scattering process can now be considered as a three-body problem, rather similar to positron scattering by atomic hydrogen but with the important difference that, because the ionization energy of an alkali atom is less than the binding energy of positronium, 6.8 eV, the positronium formation channel is open even at zero positron energy. 

Section 3.1 discusses the cases where only the channels having a common asymptotic energy determine the dominant dynamics at large distances R. There, a natural choice of R should be the ordinary distance between the two fragments the whole system asymptotically separates into. Another choice of R, appropriate for studying the dynamics in the whole region 0 < R and especially powerful for three-body problems, is discussed in Section 3.2. 

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

Double photoionization in the outer shell of rare gases by a single photon is an important manifestation of electron correlations. One specific aspect which has received much attention over the years is double photoionization in the vicinity of the double-ionization threshold. On the theoretical side, this attention is due to the possibility of deriving certain threshold laws without a full solution of the complicated three-body problem of two electrons escaping the field of the remaining ion. On the experimental side, the study of threshold phenomena always provides the challenge for mastering extremely difficult experiments. 

This is the the simplest possible molecule, the hydrogen molecule ion, H2+, a known entity [1], Strictly speaking, this presents a three-body problem - two protons and an electron - which cannot be solved exactly [2]. To a good approximation, however, the protons can be taken as stationary compared to the electron (the Bom-Oppenheimer principle) and this system can be solved exactly [3]. 

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... 

A three-body problem is shown in Fig. 8-27. In this case each of the bodies exchanges heat with the other two. The heat exchange between body 1 and body 2 would be... 

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. 

This is a three-body problem, the two plates and the room, so the radiation network is shown in Fig. 8-27. From the data of the 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. 

---