Two-body problem

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A problem dealing with the motion of two material points and with masses and , respectively, moving in three-dimensional Euclidean space when acted upon by the mutual Newton attracting forces. The problem is a special case of the -body problem, which may be described by a system of ordinary differential equations of order , and has 10 independent integrals: 6 of motion of the centre of inertia, 3 of law of areas (equivalently, conservation of angular momentum) and 1 of energy conservation [1]. The two-body problem also has three Laplace integrals (one of which is independent of the preceding ones) and is completely integrable [2].

The integration of the two-body problem is more conveniently effected in a special system of coordinates, in which these integrals are employed. If the origin of the Cartesian coordinates is placed at the centre of mass and the axis is directed along the relative angular momentum vector, then the motion of the relative position vector takes place in the plane and satisfies the system


where , is the reduced mass and is the gravitational constant. The system (1) has four integrals:

which are interconnected by the relation



i.e. the orbits of the relative position vector are conical sections with parameter , major semi-axis , eccentricity , longitude of pericentre (, ), and with focus at the coordinate origin. The location of the relative positive vector on the orbit is determined by the true anomaly , counted from the direction towards the pericentre; (2) then implies that . If , three types of orbits are possible:

I) If , they are ellipses.

II) If , they are hyperbolas.

III) If , they are parabolas.

If , the motion is rectilinear. The two-body problem describes an unperturbed Kepler motion of a planet with respect to the Sun or of a satellite with respect to a planet, etc.


[1] C.L. Siegel, "Vorlesungen über Himmelmechanik" , Springer (1956)
[2] V.K. Abalakin, et al., "Handbook of celestial mechanics and astrodynamics" , Moscow (1971) (In Russian)



[a1] H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 1–3 , Gauthier-Villars (1899)
[a2] C.L. Siegel, J. Moser, "Lectures on celestial mechanics" , Springer (1971)
[a3] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a4] R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin (1978)
How to Cite This Entry:
Two-body problem. A.D. Bryuno (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098