Abstract
A review is presented of some recent developments in the study of geomet rical aspects of the dynamics of curves in the plane. An examination of general conditions necessary for global geometric conservation laws such as those of length and enclosed area leads to a class of motions mathematically equivalent to a hierarchy of integrable systems related to the Korteweg-de Vries (KdV) equation. The KdV, modified KdV, and Harry Dym hierarchies are then seen to be three equivalent views of the same underlying dynamics. It is shown that the nonlinear transformations between these systems have straightforward geometric meaning. These dynamics are found to be local approximations to the motion of vortex patches in ideal two-dimensional fluids, a result paralleling the connection between the Nonlinear Schrodinger equation and the motion of a vortex filament in space. The Hamiltonian structure of these integrable systems is recast in a form emphasizing the geometric interpretation in the language of curve motion. Applications of these results to physical systems are suggested.
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