Central Configurations—A Problem for the Twenty-first Century

What does it take to qualify as a “problem for the twenty-first century?” Obviously, the topic must be of fundamental importance, it must be difficult to solve as manifested by the number of excellent people who have tried—but failed, and non-experts should be able to appreciate the basic issues. The particular problem described here, which involves finding certain geometric configurations that arise in several places including the Newtonian N -body problem, easily satisfies these criteria. The issue is not new: a couple of centuries ago Euler and Lagrange answered the question for the three-body problem. But the full problem for N ≥ 4 has proved resistant to solution. What makes the topic so important is that even partial answers add to our understanding of N -body systems. These brief comments already suggest why Smale [17] listed one of the problems described below as a mathematical challenge for the twenty-first century. Of particular interest, this topic is one where non-experts can be expected to add new insights. In fact, beyond describing aspects of this fascinating issue, a goal of this paper to enlist new people to this area. The general objective is to find all of the central configurations and their basic properties. While the term may be new for you, you probably have seen some of them if your interests include paddling a canoe, or watching the weather channel, as these are the configurations that tend to be formed by the swirly vortices—or cyclones in the Indian Ocean— as they move along. My examples from celestial mechanics (the mathematical study of how N -astronomical bodies move when governed by Newton’s equations) will indicate why these configurations are so important. To motivate the definition, start with the equations of motion for the Newtonian N -body problem mjr j = ∂U ∂rj = ∑