N-body problem

In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets and the visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is considerably more difficult to solve.

The classical physical problem can be informally stated as: given the quasi-steady orbital properties (instantaneous position, velocity and time)of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.

To this purpose the two-body problem has been completely solved and is discussed below; as is the famous restricted 3-Body Problem.

Knowing three orbital positions of a planet's orbit – positions obtained by Sir Isaac Newton (1643-1727) from astronomer John Flamsteed – Newton was able to produce an equation by straightforward analytical geometry, to predict a planet's motion; i.e., to give its orbital properties: position, orbital diameter, period and orbital velocity. Having done so, he and others soon discovered over the course of a few years, those equations of motion did not predict some orbits very well or even correctly. Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits.

The above discovery goes right to the heart of the matter as to what exactly the 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. Thus came the awareness and rise of the n-body "problem" in the early 17th century. These gravitational attractive forces do conform to Newton's Laws of Motion and to his Law of Universal Gravitation, but the many multiple (n-body) interactions have historically made any exact solution intractable. Ironically, this conformity led to the wrong approach.

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