N-body problem
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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.
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Mathematical formulation
The general n-body problem can be stated in the following way.
For each body i, with mass mi, let ci(t) be its trajectory in 3-dimensional space, where the parameter t is interpreted as time. Then the acceleration c''(t) of each mass mi satisfies by the law of gravity:
- <math>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}.</math>
The force on each mass mi is
- <math>F_i = c_{i}(t) m_i.\,</math>
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. 23 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.
See also Kepler's first law of planetary motion.
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 those then there do not exist any other integrals which are algebraically independent (theorem of 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 my means of convergent series. See below for details.
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 Henri 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 Price
The problem of finding the general solution of the n-body solution was considered as being very important and challenging. Indeed in the late 1800 King Oscar of Sweden, advised by Martin Leffler, established a price for anyone who could find the solution of that problem. In case the problem could not be solved, any other important contribution to classical mechanics would then considered to be prize worthy. The prize was finally awareded to Poincaré, even though he did not solved the original problem. (The first version of his contribution contained even 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.
Sundmans Theorem
In 1912, Finland-Swedish mathematician Karl Fritiof Sundman proved, that given initial data, which are analytical in the time and space variable and 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 neither have
- Triple collisions.
- Nor no-collisions singularities
Unfortunately the corresponding convergent series converges very slow. That is getting the value to any useful precision requires so many terms that his solution is of little practical use. Sundman result has been generalised to the case of n-bodies by Q. Wang in the 90.
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.
See also
- Many-body problem (quantum mechanics)
- Euler's three-body problem
- Virial theorem
- Few-body systems
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
- More detailed information on the three-body problem
- Regular Keplerian motions in classical many-body systems
- Applets demonstrating many different three-body motionsda:Trelegemeproblemet
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