International Journal of Mathematics and Mathematical Sciences
Volume 28 (2001), Issue 12, Pages 689-732

On the geometry and behavior of n-body motions

Eldar Straume

Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim N-7491, Norway

Received 16 February 2001

Copyright © 2001 Eldar Straume. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The kinematic separation of size, shape, and orientation of n-body systems is investigated together with specific issues concerning the dynamics of classical n-body motions. A central topic is the asymptotic behavior of general collisions, extending the early work of Siegel, Wintner, and more recently Saari. In particular, asymptotic formulas for the derivatives of any order of the basic kinematic quantities are included. The kinematic Riemannian metric on the congruence and shape moduli spaces are introduced via O(3)-equivariant geometry. For n=3, a classical geometrization procedure is explicitly carried out for planary 3-body motions, reducing them to solutions of a rather simple system of geodesic equations in the 3-dimensional congruence space. The paper is largely expository and various known results on classical n-body motions are surveyed in our more geometrical setting.