# N-body problem

The correct title of this article is n-body problem. The initial letter is capitalized due to technical restrictions. 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.

## 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:

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

The solutions of this system of differential equations give the positions as a function of time.

The force on each mass mi is

Fi = ci''(t)mi

## 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.

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.

## Three-body problem

File:N-body problem (3).gif
The chaotic movement of 3 interacting particles

The three-body problem is much more complicated; its solution can be chaotic. In general, the three-body problem cannot be solved analytically (i.e. in terms of a closed-form solution of known constants and elementary functions), 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 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.

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.