| his thesis explores the application of geometric mechanics to problems in 2D, incompressible, inviscid fluid mechanics. The main motivation is to try to develop symplectic integration algorithms to model the Hamiltonian structure of inviscid fluid flow. The main manifestation of this Hamiltonian or conservative nature is the preservation of the infinite family of Casimirs parametrized by the body integrals of vorticity in the 2D case, The main difficulties encountered in trying to model the Hamiltonian structure of a fluid mechanical system are that the configuration space for the Hamiltonian flow is an infinite dimensional Frechet space and that the phase space is not symplectic but Lie-Poisson. Therefore, an appropriate finite mode truncation must be constructed under the constraint that it too remains Poisson and in some sense converges to the infinite dimensional parent manifold. With such a truncation in hand, there still remains the obstacle of non-symplectic structure. This geometry invalidates the application of traditional symplectic integrators and requires a more sophisticated algorithm.;We develop a Lie-Poisson truncation on the Lie group SU(N) for the Euler equations on the special geometry of a twice periodic domain in... |
