Geodesic
The axisymmetric problem of gravitational interaction of $n$ bodies
Matematičeskoe modelirovanie, Tome 15 (2003) no. 5, pp. 27-36

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It is considered the gravitational interaction of $n$ bodies, which axially symmetric located on a plane. The force of action on each body is calculated. It is shown, that force is directed to a center of masses and is inverse to a distance to it. The expressions for a velocity, trajectory and time of moving on trajectory are received for all possible cases of interaction. The obtained exact solutions are confirmed by numerical solutions of this task in the specific case of four bodies. The exact solution gives the answers to a many of problems, which cannot be obtained by approximate solutions. Besides it allows to test approximate methods of a solution of a many bodies problem. 
@article{MM_2003_15_5_a3,
     author = {I. I. Smul'skii},
     title = {The axisymmetric problem of gravitational interaction of $n$ bodies},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {27--36},
     publisher = {mathdoc},
     volume = {15},
     number = {5},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2003_15_5_a3/}
}
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I. I. Smul'skii. The axisymmetric problem of gravitational interaction of $n$ bodies. Matematičeskoe modelirovanie, Tome 15 (2003) no. 5, pp. 27-36. http://geodesic.mathdoc.fr/item/MM_2003_15_5_a3/