N-body problem
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- The title of this article is incorrect because of technical limitations. The correct title is n-body problem. The letter n is mathematical notation and should be italicized and set in lower-case.
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 solutions of this system of differential equations give the positions as a function of time.
The force on each mass mi is
- <math>F_i = c_{i}''(t) m_i<math>
Two-body problem
Main article: 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 less 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.
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 cannot be solved analytically, although approximate solutions can be calculated by numerical methods or perturbation methods.
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 Lagrange points.
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. Poincare's work on the restricted three-body problem was the foundation of deterministic chaos. In the circular problem, there exist five equilibrium points. Three are colinear with the masses (in the rotating frame) and are unstable. The remaining two are located 60 degrees ahead of and behind the smaller mass (e.g., Jupiter) in its orbit about the larger mass (e.g., Sun), forming two equilateral triangles. 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.
In 1912, Finland-Swedish mathematician Karl Fritiof Sundman developed a convergent infinite series that provides a solution to the restricted three-body problem. Unfortunately, getting the value to any useful precision requires so many terms (on the order of 108,000,000) that his solution is of little practical use.
See also
External links
- More detailed information on the three-body problem (http://www.geom.umn.edu/~megraw/CR3BP_html/cr3bp.html)
- Regular Keplerian motions in classical many-body systems (http://www.ifmo.ru/butikov/ManyBody.pdf)
- Applets demonstrating many different three-body motions (http://www.ifmo.ru/butikov/Projects/Collection.html)de:Dreikörperproblem