Nonlinear Stability in Fluids and Plasmas

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Cartan-Hannay-Berry Phases and Symmetry

We give a systematic treatment of the treatment of the classical Hannay-Berry phases for mechanical systems in terms of the holonomy of naturally constructed connections on bundles associated to the system. We make the costructions using symmetry and reduction and, for moving systems, we use the Cartan connection. These ideas are woven with the idea of Montgomery [1988] on the averaging of connections to produce the Hannay-Berry connection

The geometry and analysis of the averaged Euler equations and a new diffeomorphism group

We present a geometric analysis of the incompressible averaged Euler equations for an ideal inviscid fluid. We show that solutions of these equations are geodesics on the volume-preserving diffeomorphism group of a new weak right invariant pseudo metric. We prove that for precompact open subsets of ${\mathbb R}^n$, this system of PDEs with Dirichlet boundary conditions are well-posed for initial data in the Hilbert space $H^s$, $s>n/2+1$. We then use a nonlinear Trotter product formula to prove that solutions of the averaged Euler equations are a regular limit of solutions to the averaged Navier-Stokes equations in the limit of zero viscosity. This system of PDEs is also the model for second-grade non-Newtonian fluids

Reduction, Symmetry and Phases in Mechanics

Various holonomy phenomena are shown to be instances of the reconstruction procedure for mechanical systems with symmetry. We systematically exploit this point of view for fixed systems (for example with controls on the internal, or reduced, variables) and for slowly moving systems in an adiabatic context. For the latter, we obtain the phases as the holonomy for a connection which synthesizes the Cartan connection for moving mechanical systems with the Hannay-Berry connection for integrable systems. This synthesis allows one to treat in a natural way examples like the ball in the slowly rotating hoop and also non-integrable mechanical systems

Induced Dirac structures on isotropy-type manifolds

A new method of singular reduction is extended from Poisson to Dirac manifolds. Then it is shown that the Dirac structures on the strata of the quotient coincide with those of the only other known singular Dirac reduction metho