Problem with NR body

The problem with NR body consists in nellement solving the equations of the movement of Newton of $N$ body interacting Gravitation, knowing their masses like their positions and speeds initial.

It is about a fundamental mathematical problem for the traditional Astronomie, i.e. if the effects of the theory of the General relativity of Einstein can be neglected: small speeds of the bodies in front of speed of light in the vacuum, and weak fields of gravitation, which is primarily the case in our Solar system.

The problem with NR body also arises within the framework of general relativity; its study is even more difficult there than within the Newtonian framework.

Mathematical formulation

$q_j (0), \ quad \ dowry q_j (0), j=1, \ ldots, N$ with $q_j (0) \ neq q_k (0)$

m_j \ ddot q_j = \ gamma \ sum \ limits_ {K \ neq J} ^ {N} \ frac {m_j m_k (q_j-q_k)}

Problem with two bodies

First triumph of the mechanics of Newton, the Problem with two bodies, is entirely soluble analytically Keplerian Mouvement: it is said that it is about an integrable problem . All the students of first cycle in physics discover one day the wheels of them.

Problem with NR body

Apart from some extremely rare cases where an exact solution is known, it is in general necessary to resort to methods of approximate resolutions . Two approaches are used:

the Theory of the disturbances, which makes it possible to make analytical calculations approximate in the form of developments in series.

the numerical Analysis. In Programming, the problem of the simulation of $N$ body should be theoretically of order $N^2$ , because all the interactions of body two to two should be considered a priori. Considerations of recursive space cutting (See: Algorithme of Barnes-Hut) however makes it possible to only arrive at very correct approximations in a time about $N \ log N$ .

Notice on the problem with three bodies

Contrary to a spread idea, the problem with three bodies has an exact analytical solution, discovered by Karl Sundman in 1909. Unfortunately, this solution is appeared as a infinite series which converges very slowly, which makes in practice it useless to make predictions in a reasonable time.

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