Symplectic Reduction for Semidirect Products and Central Extensions

This paper proves a symplectic reduction by stages theorem in the context of geometric mechanics on symplectic manifolds with symmetry groups that are group extensions. We relate the work to the semidirect product reduction theory developed in the 1980's by Marsden, Ratiu, Weinstein, Guillemin and Sternberg as well as some more recent results and we recall how semidirect product reduction finds use in examples, such as the dynamics of an underwater vehicle. We shall start with the classical cases of commuting reduction (first appearing in Marsden and Weinstein, 1974) and present a new proof and approach to semidirect product theory. We shall then give an idea of how the more general theory of group extensions proceeds (the details of which are given in Marsden, Misiołek, Perlmutter and Ratiu, 1998). The case of central extensions is illustrated in this paper with the example of the Heisenberg group. The theory, however, applies to many other interesting examples such as the Bott-Virasoro group and the KdV equation

Geometric Analysis of the Generalized Surface Quasi-Geostrophic Equations

We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach