Three-body problem, general

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

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... 

As this book shows, TST is becoming more and more of a multidisciplinary endeavor. Some new applications may be found outside the realm of chemical physics. In particular, Marsden and co-workers [13] applied those concepts to celestial mechanics of small bodies, while the general theory of the Keplerian three-body problem makes use of TST, even if in a highly singular case [14,15]. Much work remains to be done outside of chemical physics types of dynamics. I come back to this question in the conclusion. 

The quantum treatment of this molecule appears to be a straightforward extension of that for the hydrogen atom (see Section 5.1 for the solutions). One more proton has been added, and symmetry of the system has changed from spherical to cylindrical. But this apparently small change has converted the system into the so-called three-body problem, for which there is no general solution, even in classical physics. Fortunately, there is an excellent approximation that treats the nuclear and electronic motions almost as if they were independent. This method allows us to solve the Schrodinger equation exactly for the electron in Hj and provides an approximate solution for the protons as if they were almost independent. 

Exploiting a four-dimensional rotation group analysis, the transformation between harmonic expansions in the two coordinates systems was given explicitly [32], as well as the most general representation in terms of Jacobi functions [2], In practice, however, the two representations are in one form or another those being used in all applications and specifically in recent treatments of the elementary chemical reactions as a three-body problem [11,33-36]. For example, Eqs. (29)-(31) and Eqs. (47)-(49) permitted to establish [37] the explicit connection between coordinates for entrance and exit channels to be used in sudden approximation treatments of chemical reactions [38],... 

The general three-body problem Three bodies with finite masses moving under their gravitational attraction. This is a model for a triple stellar system. In many astronomical applications one of the three bodies has a large mass and the other two bodies have small, but not negligible masses. This is a model for an extrasolar planetary system, or a system of two satellites moving around a major planet. In the latter two cases the two small bodies move in perturbed Keplerian orbits. 

Most numerical exploration of periodic solutions have in fact been carried out in the restricted problem of three bodies, the simplified version in which the mass, m2, of the body P2 is considered to be too small for it to influence the motion of the bodies l ) and Pi, but remains fully influenced by the gravitational attraction of them. This system can be reached from the general three-body problem by a limiting process in which m2 tends to zero, after the factor m2 has been cancelled from the equation of motion for P2. But it is more usual for its equations of motion to be derived from first principles. It possesses a single integral of motion, Jacobi s integral, which is usually derived from first principles from its... 

In this way we are led to the family of periodic solutions of the general three-body problem, of Poincare s first sort, except when the mean motions are in the ratio of two successive integers. With this exception (whose significance we will see later), then, the process leads, in the... 

Abstract Three-body systems with Hill-type stability are the generalization to the general three-body problem of the Hill-stable orbits of the circular restricted three-body problem. 

L5 the three-body system of interest cannot approach equilateral configurations. Furthermore if 1Jp/a is larger than the value of q/v at the saddle points the zone of possible motion becomes disconnected into two or three parts and we reach then the extension of Hill stability of the circular restricted three-body problem to the general three-body problem. 

In spite of a greater simplicity than the general three-body problem, and especially in spite of the absence of motions of exchange type, the three-body systems with Hill type stability have a wide variety of solutions and, when the mutual inclination is large, they undergo large perturbations that can sometimes lead to the collision of the two bodies of the binary. 

The complexity of all these results give a small taste of the far larger complexity of the general three-body problem with possible escape of any of the three bodies. It would be especially of great interest to look for motions of exchange type and to determine how close they can approach to the Hill type stability. 

Bozis G. (1976). Zero velocity surfaces for the general planar three-body problem. 

Marchal C. and Saari D.G. (1975). Hill s regions for the general three-body problem. Celestial Mechanics 12, p. 115. 

The present section excludes discussion of the two- and three-body problems, and reduces to a minimum the treatment of resonant processes, since these topics are dealt with in other articles in this Encyclopedia (Vol. XXXIX and XL). General information on neutron induced reactions is contained in references [d] and [7] and total cross sections are given by Adair and by Hughes and Harvey [3d]. The reactions observed are listed in Table 7. 

Hyperspherical coordinates were introduced by Delves [52] and the formalism of hyperspherical expansion was further developed by many authors [40,53,54] for three-body or more complicated bound states. The usefulness of this method for baryon spectroscopy was shown by several groups [55]. The basic idea is rather simple the two relative coordinates are merged into a single six-dimensional vector. The three-body problem in ordinary space becomes equivalent to a two-body problem in six dimensions, with a noncentral potential. A generalized partial wave expansion leads to an infinite set of coupled radial equations. In practice, however, a very good convergence is achieved with a few partial waves only. 

For this reason, we will restrict our subsequent approach to planar configurations of the two electrons and of the nucleus, with the polarization axis within this plane. This presents the most accurate quantum treatment of the driven three body Coulomb problem to date, valid in the entire nonrelativistic parameter range, without any adjustable parameter, and with no further approximation beyond the confinement of the accessible configuration space to two dimensions. Whilst this latter approximation certainly does restrict the generality of our model, semiclassical scaling arguments suggest that the unperturbed three... 

The hyperspherical method, from a formal viewpoint, is general and thus can be applied to any N-body Coulomb problem. Our analysis of the three body Coulomb problem exploits considerations on the symmetry of the seven-dimensional rotational group. The matrix elements which have to be calculated to set up the secular equation can be very compactly formulated. All intervals can be written in closed form as matrix elements corresponding to coupling, recoupling or transformation coefficients of hyper-angular momenta algebra. 

Since the VLPT is formally exact, instead of looking for the best construction procedure for high order RDM s, the problem is reformulated as how to estimate the pure two-, three-, four-body correlation terms in the VLPT or, equivalently, the A, A, % whose expressions we now know and which in general are closely related but are not respectively identical to the two-body, three-body and four-body correlation terms. This redefinition of the problem is more precise - and therefore valuable - although we are fully aware that it does not provide the solution by itself. 

Others (e.g., Fukashi Sasaki s upper bound on eigenvalues of 2-RDM [2]). Claude Garrod and Jerome Percus [3] formally wrote the necessary and sufficient A -representability conditions. Hans Kummer [4] provided a generalization to infinite spaces and a nice review. Independently, there were some clever practical attempts to reduce the three-body and four-body problems to a reduced two-body problem without realizing that they were actually touching the variational 2-RDM method Fritz Bopp [5] was very successful for three-electron atoms and Richard Hall and H. Post [6] for three-nucleon nuclei (if assuming a fully attractive nucleon-nucleon potential). 

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