V.33 The Three-Body Problem

The three-body problem can be simply stated: three point masses move in space under their mutual gravitational attraction; given their initial positions and velocities, determine their subsequent motion. Initially, it may come as a surprise that this is a difficult problem, since the analogous two-body problem can be solved fairly simply: more precisely, given any set of initial conditions, we can write down a formula, in terms of elementary functions (these are functions that can be built up using the basic operations of arithmetic, together with a few standard functions such as the EXPONENTIAL [III.25] and TRIGONOMETRIC [III.92] functions), that tells us the subsequent positions and velocities of the bodies. However, the three-body problem is a complicated nonlinear problem and it cannot be solved in this way, even if we are prepared to enlarge our stock of “standard functions” somewhat. NEWTON [VI.14] himself speculated that an exact solution “exceeds, if I am not mistaken, the force of any human mind,” while HILBERT [VI.63], in his celebrated Paris address of 1900, put the problem in a category similar to FERMAT’S LAST THEOREM [V.10]. The problem can be extended to any number of bodies and in the general case it is known as the n-body problem.

Recall that the gravitational force of a particle P₁ on a particle P₂ has magnitude k²m₁m₂/r² (in suitable units), where k is the Gaussian gravitational constant, particle P_i has mass m_i, and the distance between the particles is r. The direction of this force on P₂ is toward P₁ (and there is a force of the same magnitude on P₁ in the direction of P₂). Recall also Newton’s second law: force equals mass times acceleration. From these two laws we can easily derive the equations of motion for the three-body problem. Let the particles be P₁, P₂, and P₃. Write m_i for the mass of P_i, r_ij for the distance between P_i and P_j, and q_ij for the jth coordinate of the position of P_i. Then the equations of motion are

Here, i runs from 1 to 3; thus, there are nine equations, all derived from the simple laws above. For instance, the left-hand side of the first equation is the component of the acceleration of P₁ in the ith direction, and the right-hand side is the component of the force acting on P₁ in this direction, divided by m₁.

If the units are chosen so that k² = 1, then the potential energy V of the system is given by

Setting

we can rewrite the equations in the HAMILTONIAN FORM [IV.16 §2.1.3]

which is a set of eighteen first-order differential equations. Since this set is easier to use, it is now generally preferred to (1).

A standard way of decreasing the complexity of a system of differential equations is to find an algebraic integral for it: that is, a quantity that will remain constant for any given solution and that can be expressed as an integral that gives rise to an algebraic dependence between the variables. This allows us to reduce the number of variables by expressing some of them in terms of others. The three-body problem has ten independent algebraic integrals: six of them tell us about the motion of the center of mass (three for the position variables and three for the momentum variables), three integrals express the conservation of angular momentum, and one expresses conservation of energy. These ten independent integrals were known to EULER [VI.19] and LAGRANGE [VI.22] in the middle of the eighteenth century, and in 1887 Heinrich Bruns, professor of astronomy at Leipzig, proved that there are no others, a result sharpened by POINCARÉ [VI.61] two years later. By the use of these ten integrals, together with the “elimination of the time” and the “elimination of the nodes” (a procedure first made explicit by JACOBI [V1.35]), the original system of order eighteen can be reduced to one of order six, but it can be reduced no further. Hence, any general solution of (2) cannot be given by a simple formula: the best we can hope for is a solution in the form of an infinite series. It is not difficult to find series that work well enough for a limited time span: the problem is to find series that work for any initial configuration and for any time span, no matter how long. There is also the question of collisions. A complete solution to the problem has to take account of all possible motions of the bodies, including determining which initial conditions lead to binary and triple collisions. Since collisions are described by singularities in the differential equations, this means that to find a complete solution the singularities have to be understood.

This turns out to be a more interesting problem than one might think. It is obvious from the equations that a collision gives rise to a singularity, but it is less clear whether there can be any other kind of singular behavior. In the case of the three-body problem, the answer was supplied by Painlevé in 1897: the collisions are the only singularities. However, for more than three bodies the answer turned out to be different. In 1908 a Swedish astronomer, Hugo von Zeipel, showed that noncollision singularities can occur only if the system of particles becomes unbounded in a finite amount of time. A good example of such a singularity was found by Zhihong Xia for the five-body problem in 1992. In this case there are two pairs of bodies, the bodies in each pair having equal mass, and a fifth body with very small mass. The bodies in a pair move in very eccentric orbits parallel to the xy-plane, with the two pairs on opposite sides of this plane and rotating in opposite directions. A fifth particle is then added to the system. Its motion is confined to the z-axis and oscillates between the two pairs. Xia showed that the motion of the fifth particle forced the two pairs to move away from the xy-plane, but that it also came closer and closer to colliding with the pairs, giving it larger and larger bursts of acceleration, and that as this happened the two pairs were forced out to infinity in finite time.

As well as trying to solve the problem in general, one can look for interesting particular solutions. A central configuration is defined to be a solution in which the geometric configuration remains constant. The first examples were discovered by Euler in 1767: they were solutions in which the bodies always lie on a straight line and revolve with uniform angular velocity in circles or ellipses about their common center of mass. In 1772 Lagrange discovered solutions in which the bodies are always at the vertices of an equilateral triangle that rotates uniformly about the center of mass. For almost all sets of initial conditions for these solutions, the size of the triangle changes as it rotates so that each body describes an ellipse.

However, despite the discovery of the particular solutions and a century of unrelenting work on the problem, the mathematicians of the nineteenth century were unable to find a general solution. Indeed, the problem was considered so hard that in 1890 Poincaré was led to declare that he thought it impossible without the discovery of some significant new mathematics. But, contrary to Poincaré’s expectation, less than twenty years later a young Finnish mathematical astronomer, Karl Sundman, using only existing mathematical techniques, astonished the mathematical world by obtaining uniformly convergent infinite series that mathematically “solved” the problem. Sundman’s series, which are in powers of t^1/3, are convergent for all real t, except for the negligible set of initial conditions for which the angular momentum is zero. To deal with binary collisions, Sundman used the technique of regularization, or analytically extending a solution beyond the collision, but he was unable to deal with triple collisions because in order for such a collision to occur the angular momentum must be zero.

Although it was a remarkable mathematical achievement, Sundman’s solution leaves many questions unanswered. It provides no qualitative information about the behavior of the system and, worse, because the series converges so slowly it is of no practical use. To determine the motion of the bodies for any reasonable period of time would require the summation of something of the order of 10^8000000 terms, a calculation that is patently unrealistic. Thus, Sundman left plenty still to do, and work on the problem (and the related n-body problem) has continued up to the present day, with exciting results continuing to appear. One recent example is a convergent power-series solution for the general n-body problem, which was discovered by Don Wang in 1991.

Since the three-body problem itself proved so intractable, simplified versions were developed, of which the most famous is the one now known as the restricted three-body problem (the name is due to Poincaré), which was first investigated by Euler. In this case, two of the bodies (the primaries) revolve around their joint center of mass in circular orbits under the influence of their mutual gravitational attraction, while the third body (the planetoid), which is assumed to have such small mass that the force it exerts on the other two bodies can be neglected, moves in the plane defined by the primaries. The advantage of this formulation is that the motion of the primaries can be treated as a two-body problem and is hence known; it remains only to investigate the motion of the planetoid, which can be done using perturbation theory. Although the restricted formulation might appear artificial, it provides a good approximation to real physical situations, such as, for example, the problem of determining the motion of the Moon around Earth given the presence of the Sun. Poincaré wrote extensively on the restricted problem, and the techniques he developed to tackle it led to his discovery of mathematical chaos, as well as laying the foundations for modern DYNAMICAL SYSTEMS [IV.14] theory.

Apart from its intrinsic appeal as a problem that is simple to state, the three-body problem has a further attribute that has contributed to its attraction for potential solvers: its intimate link with the fundamental question of the stability of the solar system. That is the question of whether the planetary system will always keep the same form as it has now, or whether, eventually, one of the planets will escape or, perhaps worse, experience a collision. Since bodies in the solar system are approximately spherical and their dimensions extremely small when compared with the distances between them, they can be considered as point masses. Ignoring all other forces, such as solar winds or relativistic effects, and taking only gravitational forces into account, the solar system can be modeled as a ten-body problem with one large mass and nine small ones, and it can be investigated accordingly.

Over the years, attempts to find a solution to the three-body problem (and the related n-body problem), have spawned a wealth of research. As a result, the importance of the problem is as much in the mathematical advances it has generated as in the problem itself. A notable example of this is the development of KAM theory, which provides methods for integrating perturbed Hamiltonian systems and obtaining results that are valid for infinite periods of time. This was developed in the 1950s and 1960s by KOLMOGOROV [VI.88], Arnold, and Moser.

Thurston’s Geometrization Conjecture

See THE POINCARÉ CONJECTURE [V.25]