國立清華大學 數學系

Consider a collection of n rigid, massive bodies interacting according to their mutual gravitational attraction. A relative equilibrium motion is one where the entire configuration rotates rigidly and uniformly about a fixed axis — all of the bodies are phase locked. Such a motion is possible only for special positions and orientations of the bodies. A minimal energy motion is one which has the minimum possible energy in its fixed angular momentum level. While every minimal energy motion is a relative equilibrium motion, the main result here is that a relative equilibrium motion of n>=3 disjoint rigid bodies is never an energy minimizer. Since energy minimizers are the expected final states produced by tidal friction, phase locking of 3 or more bodies will not occur by this mechanism. For the case n=2 we work out some Morse theoretical estimates for the number of relative equilibria.