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Volume 12: Pages 197-203, 1999
The n‐Body Problem Solved in Two Dimensions
Kern E. Kenyon
4632 North Lane, Del Mar, California 92014‐4134 U.S.A.
A force‐balance method, equivalent to Newton's second law, provides a way to set up the solution of the 3‐body problem in a fixed plane that is thought to be original. For each of the three gravitating mass points that rotate about the center of mass there is a balance of two equal but opposite forces in the direction perpendicular to the orbit: the outward centrifugal force and the total inward component of the gravitational force normal to the orbit. In the direction tangent to the orbit no balance of forces is possible in general, and therefore the masses continually accelerate along their paths. After inserting a few mathematical relations, among which is a simplified form of the radius of curvature, the force balance can be written as a harmonic‐oscillator equation in the spatial (polar) coordinates of the given mass, on whose right‐hand side is a fully nonlinear coefficient involving the velocity and coordinates of that mass as well as the masses and coordinates of the other two masses. The nonlinear coefficient is written down explicitly in terms of elementary functions, and it contains two coupling terms, each of which consists of functions of the coordinates of one pair of masses only. Altogether there are three separate harmonic‐oscillator equations, one for each mass, to be solved simultaneously for the three orbits. Because of the nonlinearity the solutions of the orbit equations must usually be carried out by a numerical integration, after specifying the initial positions and velocities of the three masses. No numerical solutions are attempted here. The orbital solution to the n‐body problem in two dimensions is set up analogously to that of the 3‐body problem. There will be n harmonic‐oscillator equations to be solved simultaneously by numerical integration for the orbits of the n masses, once the initial conditions are given. Each nonlinear coefficient is more bulky now because it contains n − 1 coupling terms, compared to two for the 3‐body problem, but this is only a difficulty of degree, not one of kind.
Keywords: 3‐body problem, n‐body problem, two‐dimensional solution
Received: March 19, 1997; Published online: December 15, 2008