Curve minimization problems, Lie algebras of Hamiltonian functions, the brachistochrone, the Reeds-Shepp car in two and three dimensions, spinning tops, and elastic rods

This lecture will be given in lieu of Velimir Jurdjevic's talk, which was scheduled for September 19 but had to be cancelled. The talk will outline a general program of research in which the speaker and Jurdjevic have been involved for many years, which began with their joint papers in the 1970's and was since then pursued by both authors by following somewhat separate but closely related directions. The purpose of this work is to relate in a systematic way the properties of the solutions of various curve minimization problems to the structure of certain Lie algebras of Hamiltonian functions or, more precisely, of certain pairs (L,F) consisting of a Lie algebra L of Hamiltonian functions together with a parameterized family F of generators of L. It will be explained how these Lie algebras arise naturally in the study of the minimization problems. Examples of two kinds of problems will be presented. First, simple examples where the solutions can be found directly, and which, precisely because of their simplicity, can be used to illustrate how the general theory works. Second, examples of much harder problems for which the theory yields nontrivial and sometimes surprising new results. The classical brachistochrone problem will be discussed as an example lying somewhere in between these two categories, and shown to be related to the diamond Lie algebra. Included in the discussion of the second kind of examples will be the three-dimensional case of the so-called "Reeds-Shepp car," which remained open until it was solved using our methods and, possibly, Jurdjevic's results on the Kowalewska case of the spinning top and its relation with the equations for elastic rods.