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CHAPTER XIII - THE REDUCTION OF THE PROBLEM OF THREE BODIES

Published online by Cambridge University Press:  05 August 2012

E. T. Whittaker
Foreword by
Sir William McCrae
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Summary

Introduction.

The most celebrated of all dynamical problems is known as the Problem of Three Bodies, and may be enunciated as follows:

Three particles attract each other according to the Newtonian law, so that between each pair of particles there is an attractive force which is proportional to the product of the masses of the particles and the inverse square of their distance apart: they are free to move in space, and are initially supposed to be moving in any given manner; to determine their subsequent motion.

The practical importance of this problem arises from its applications to Celestial Mechanics: the bodies which constitute the solar system attract each other according to the Newtonian law, and (as they have approximately the form of spheres, whose dimensions are very small compared with the distances which separate them) it is usual to consider the problem of determining their motion in an ideal form, in which the bodies are replaced by particles of masses equal to the masses of the respective bodies and occupying the positions of their centres of gravity.

The problem of three bodies cannot be solved in finite terms by means of any of the functions at present known to analysis. This difficulty has stimulated research to such an extent, that since the year 1750 over 800 memoirs, many of them bearing the names of the greatest mathematicians, have been published on the subject.

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A Treatise on the Analytical Dynamics of Particles and Rigid Bodies , pp. 339 - 357
Publisher: Cambridge University Press
Print publication year: 1988

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