Abstract
The well-known central configurations of the three-body problem give rise to periodic solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that such rigid motions, called relative equilibrium solutions, are the only solutions with constant moment of inertia. This result will be proved here for the Newtonian three-body problem in $\R^d$ with three positive masses. The proof makes use of some computational algebra and geometry. When $d\le 3$, the rigid motions are the planar, periodic solutions arising from the five central configurations, but for $d\ge 4$ there are other possibilities.
Highlights
It is a well-known property of the Newtonian n-body problem that the center of mass of the bodies moves along a line with constant velocity
The goal of this paper is to provide a proof for the three-body problem in Rd
Note that for the three-body problem, the initial position and velocity vectors of any solution with center of mass at the origin will always span a subspace of dimension d ≤ 4 and the solution will remain in this subspace for all time
Summary
It is a well-known property of the Newtonian n-body problem that the center of mass of the bodies moves along a line with constant velocity. The corresponding solutions rotate in a plane containing the masses with constant angular velocity. Note that for the three-body problem, the initial position and velocity vectors of any solution with center of mass at the origin will always span a subspace of dimension d ≤ 4 and the solution will remain in this subspace for all time. A solution of the three-body problem has constant moment of inertia if and only if it is a relative equilibrium solution. Consider a solution with constant moment of inertia To show that it is a rigid motion, it suffices to show that the mutual distances rij are constant. Using the assumption of constant moment of inertia, one can derive a set of algebraic equations involving the variable rij and certain velocity variables. A Mathematica notebook presenting details of all computations with extensive comments can be found at [10]
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