Curvature and Instability

The Euler equations of a rigid body can be understood as the geodesic equations for a metric on the rotation group. A rapid introduction to the Riemannian geometry of Lie groups (following Milnor) is given and illuminated by the example of the rigid body. The deep generalization of Arnold to the case of an incompressible fluid is then explained. The Euler equations of an ideal incompressible fluid are shown to be geodesics of the group of volume preserving diffeomorphisms. The curvature of this metric is calculated. Contrary to the case of the rigid body, the curvature is negative, implying that the dynamics of such a fluid is highly unstable. Some ideas on how geodesic dynamics is modified by dissipation are introduced. This leads to new generalizations of Riemannian geometry.