History of Maxwell's problem in the theory of optimization (Q2713219)

The author begins with the work for which Maxwell won the Adams Prize in 1859, in which he showed that a solid ring around a planet could be stable only if its mass distribution were so nonuniform as to approximate a satellite, and that a liquid ring would be destroyed by resonant waves. (Laplace had used a fictitious liquid layer covering the ring to study its cross-sectional shape.) It followed that Saturn's rings must consist of meteorites, a fact which was confirmed by the Pioneer photographs. Maxwell's problem, as stated by the author, is to develop an algorithm to determine the stability of a motion described by a system of ordinary differential equations. The evolution of this problem, and the contributions to it by Routh, Hurwitz, and others are described.