Discrete Euler-Poincare Equations and the SU(N)-\alpha Model of 2D Hydrodynamics

Sergey Pekarsky, Control and Dynamical Sytems, California Institute of Technology

Discrete analogues of Euler-Poincar\'{e} and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups $G$ with Lagrangians ${\mathcal L}:TG \rightarrow {\mathbb R}$ that are $G$-invariant. These discrete equations provide ``reduced'' numerical algorithms which manifestly preserve the symplectic structure.

A comparison between Euler and Euler-alpha models of $2 D$ hydrodynamics on a torus using the finite dimentional approximation to the diffeomorphism group of ${\mathbb T}^2$ is done. This approximation is constructed using the su$(N)$ algebra and relies on the fact that for $N \rightarrow \infty$ su$(N)$ limits to the Lie algebra of the diffeomorphism group. The resulting models are Lie-Poisson systems on SU$(N)$ equipped with different metrics.