Geometrical Aspects in Hydrodynamics and Integrable Systems

The motion of fluid particles of an inviscid incompressible fluid on a bounded domain is formulated from a Lagrangian point of view. This is accomplished by observing that Euler's equation of motion is a geodesic equation on a group of volume-preserving diffeomorphisms with the metric defined by the kinetic energy. This formulation is based on Riemannian geometry and Lie group theory, first developed by Arnold (1966). Behaviors of the geodesics are characterized by Riemannian (sectional) curvatures, which are shown to be mostly negative (with some exceptions). This property is related to the mixing and ergodicity of the fluid motions. Free rotation of a rigid body fixed at a point gives a simplest example of the dynamical systems which are integrable and represented with such formulation. The same method is applied to the other integrable systems such as the vortex-filament equation or the KdV equation. In contrast to the hydrodynamic system, sectional curvatures are found to be mostly positive (with exceptions). Thus it is found that integrable systems are more stable in the behavior of geodesics than the hydrodynamic system governed by the Euler's equation of motion. Received 16 January 1997 and accepted 30 May 1997