10 Facts About N-body problem

In physics, the -body N-body problem is the N-body problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally.

Two-body N-body problem has been completely solved and is discussed below, as well as the famous restricted three-body N-body problem.

Above discovery goes right to the heart of the matter as to what exactly the -body N-body problem is physically: as Newton realized, it is not sufficient to just specify the initial position and velocity, or three orbital positions either, to determine a planet's true orbit: the gravitational interactive forces have to be known too.

Newton does not say it directly but implies in his Principia the -body N-body problem is unsolvable because of those gravitational interactive forces.

The problem as stated originally was finally solved by Karl Fritiof Sundman for and generalized to by L K Babadzanjanz and Qiudong Wang.

Hamilton's equations show that the -body N-body problem is a system of first-order differential equations, with initial conditions as initial position coordinates and initial momentum values.

Two-body N-body problem was completely solved by Johann Bernoulli by classical theory by assuming the main point-mass was fixed; this is outlined here.

The fixed point for two isolated gravitationally interacting bodies is their mutual barycenter, and this two-body N-body problem can be solved exactly, such as using Jacobi coordinates relative to the barycenter.

Many earlier attempts to understand the Three-body N-body problem were quantitative, aiming at finding explicit solutions for special situations.

Planetary N-body problem is the -body N-body problem in the case that one of the masses is much larger than all the others.