N-body problem

The correct title of this article is n-body problem. The initial letter is capitalized due to technical restrictions, and the "n" should be italicized since it is used as mathematical notation.

The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i.e., Newton's laws of motion and Newton's law of gravity.

Mathematical formulation

The general n-body problem can be stated in the following way.

For each body i, with mass m_i, let c_i(t) be its trajectory in 3-dimensional space, where the parameter t is interpreted as time; $\gamma$ denotes the gravitational constant. Then the acceleration c(t) of each mass mi'' satisfies by the law of gravity:

c_{i}''(t) = \gamma \sum_{1 \le j \le n, i \ne j} m_j \frac{c_j(t) - c_i(t)}{\|c_j(t) - c_i(t)\|^3}.

The force on each mass m_i is

F_i = c_{i}''(t) m_i.\,

The solutions of this system of differential equations give the positions as a function of time. The goal is then to find and understand all physical solutions to these equations.

Two-body problem

If the common center of mass of the two bodies is considered to be at rest, each body travels along a conic section which has a focus at the centre of mass of the system (in the case of a hyperbola: the branch at the side of that focus).

If the two bodies are bound together, they will both trace out ellipses; the potential energy relative to being far apart (always a negative value) has an absolute value greater than the total kinetic energy of the system; the sum of both energies is negative. (Energy of rotation of the bodies about their axes is not counted here).

If they are moving apart, they will both follow parabolas or hyperbolas.

In the case of a hyperbola, the potential energy has an absolute value smaller than the total kinetic energy of the system; the sum of both energies is positive.

In the case of a parabola, the sum of both energies is zero. The velocities tend to zero when the bodies get far apart.

Note: The fact that a parabolic orbit has zero energy arises from the assumption that the gravitational potential energy goes to zero as the bodies get infinitely far apart. One could assign any value (e.g. 42 joules) to the potential energy in the state of infinite separation. That state is assumed to have zero potential energy (i.e. 0 joules) by convention.

Three-body problem

The three-body problem is much more complicated; its solution can be chaotic. In general, the three-body problem (and the n-body problem for n>3) cannot be solved by the method of first integrals. That is for the 18 integrals only 10 can be solved by the conservation laws. Besides these 10 integrals there do not exist any other integrals which are algebraically independent (a theorem of Heinrich Bruns, which was generalised by Poincaré). These results however do not imply that there does not exist a general solution of the n-body problem or that the perturbation series (Linstedt series) diverges. Indeed Sundman provided such a solution by means of convergent series. See below for details.

A major study of the Earth-Moon-Sun system was undertaken by Charles Delaunay, who published two volumes on the topic, each of 900 pages in length, in 1860 and 1867. Among many other accomplishments, the work already hints at chaos, and clearly demonstrates the problem of so-called "small denominators" in perturbation theory.

Euler's three-body problem

The restricted three-body problem assumes that the mass of one of the bodies is negligible; the circular restricted three-body problem is the special case in which two of the bodies are in circular orbits (approximated by the Sun - Earth - Moon system). For a discussion of the case where the negligible body is a satellite of the body of lesser mass, see Hill sphere; for binary systems, see Roche lobe; for another stable system, see Lagrangian point.

The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, notably Lagrange in the 18th century and Poincaré in at the end of the 19th century. Poincaré's work on the restricted three-body problem was the foundation of deterministic chaos theory. In the circular problem, there exist five equilibrium points. Three are collinear with the masses (in the rotating frame) and are unstable. The remaining two are located on the third vertex of both equilateral triangles of which the two bodies are the first and second vertices. This may easier to visualize if one considers the more massive body (e.g., Sun) to be "stationary" in space, and the less massive body (e.g., Jupiter) to orbit around it, with the Lagrangian points maintaining the 60 degree-spacing ahead of and behind the less massive body in its orbit (although in reality neither of the bodies is truly stationary; they both orbit the center of mass of the whole system). For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrange points.

King Oscar Prize

The problem of finding the general solution of the n-body solution was considered very important and challenging. Indeed in the late 1800s King Oscar of Sweden, advised by Martin Leffler, established a prize for anyone who could find the solution to the problem. In case the problem could not be solved, any other important contribution to classical mechanics would then considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. (The first version of his contribution even contained a serious error; for details see the article by Diacu.) The version finally printed contained many important ideas which lead to the theory of chaos. The problem as stated originally was finally solved by Karl Sundman.

Sundman's theorem

In 1912, Finland-Swedish mathematician Karl Fritiof Sundman proved that, given initial data which satisfy certain restrictions, there exists a global in time solution, which is analytical in the time and space variables. The condition on the initial data are such that the corresponding solutions will not lead to triple collisions. It was later shown by Saari, that these collisions are not generic in some technical sense.

Unfortunately the corresponding convergent series converges very slowly. That is, getting the value to any useful precision requires so many terms that his solution is of little practical use. Sundman's result was generalised to the case of n bodies by Q. Wang in the 1990s. He however had to assume that the initial conditions are such that no no-collisions singularities occur. Painlevé showed at the end of the 19th century that such singularities cannot occur in the 3-body problem and conjectured that they will be present for n>3.

Trivia

The three-body problem is figured prominently in the Criminal Minds television series episode "Compulsion."
The n-body problem also appears on the 1951 science fiction movie The Day the Earth Stood Still, where Klaatu solves it in order to attract a scientist's attention.

References

Diacu, F.: The solution of the n-body Problem, The Mathematical Intelligencer,1996,18,p.66--70
Mittag-Leffler, G.: The n-body problem (Price Announcement), Acta Matematica, 1985/1986,7
Saari, D.: A visit to the Newtonian n-body Problem via Elementary Complex Variables, American Mathematical Monthly, 1990, 89, 105-119
Wang, Q. The global solution of the n-body problem,Celestial Mechanics, 1991, 50,73-88
Sundman, K. E.: ''Memoire sur le probleme de trois corps, Acta Mathematica 36 (1912): 105--179.
Hagihara, Y: Celestial Mechanics. (Vol I and Vol II pt 1 and Vol II pt 2.) MIT Press, 1970.

Boccaletti, D. and Pucacco, G.: Theory of Orbits (two volumes). Springer-Verlag, 1998.

External links

Celestial mechanics | Classical mechanics | Problems

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