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Measurement of extremely low fluid permeabilities of rocks significant to studies of the origin of life Final report
Permeater for measuring low fluid permeabilities of rocks used to study origin of lif
Gauge Theory for Finite-Dimensional Dynamical Systems
Gauge theory is a well-established concept in quantum physics,
electrodynamics, and cosmology. This theory has recently proliferated into new
areas, such as mechanics and astrodynamics. In this paper, we discuss a few
applications of gauge theory in finite-dimensional dynamical systems with
implications to numerical integration of differential equations. We distinguish
between rescriptive and descriptive gauge symmetry. Rescriptive gauge symmetry
is, in essence, re-scaling of the independent variable, while descriptive gauge
symmetry is a Yang-Mills-like transformation of the velocity vector field,
adapted to finite-dimensional systems. We show that a simple gauge
transformation of multiple harmonic oscillators driven by chaotic processes can
render an apparently "disordered" flow into a regular dynamical process, and
that there exists a remarkable connection between gauge transformations and
reduction theory of ordinary differential equations. Throughout the discussion,
we demonstrate the main ideas by considering examples from diverse engineering
and scientific fields, including quantum mechanics, chemistry, rigid-body
dynamics and information theory
Stability of Relative Equilibria of Point Vortices on a Sphere and Symplectic Integrators
This paper analyzes the dynamics of N point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are provided for the (integrable) case N = 3. Stability of relative equilibria is analyzed by the energy-momentum method. Explicit criteria for stability of different configurations with generic and non-generic momenta are obtained. In each case, a group of transformations is specied, such that motion in the original (unreduced) phase space is stable modulo this group. Finally, we outline the construction of a symplectic-momentum integrator for vortex dynamics on a sphere
Resonant Geometric Phases for Soliton Equations
The goal of the present paper is to introduce a multidimensional generalization of asymptotic reduction given in a paper by Alber and Marsden [1992], to use this to obtain a new class of solutions that we call resonant solitons, and to study the corresponding geometric phases. The term "resonant solitons" is used because those solutions correspond to a spectrum with multiple points, and they also represent a dividing solution between two different types of solitons. In this sense, these new solutions are degenerate and, as such, will be considered as singular points in the moduli space of solitons
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 , this system of PDEs with Dirichlet boundary conditions are
well-posed for initial data in the Hilbert space , . 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
Cocycles, compatibility, and Poisson brackets for complex fluids
Motivated by Poisson structures for complex fluids containing cocycles, such
as the Poisson structure for spin glasses given by Holm and Kupershmidt in
1988, we investigate a general construction of Poisson brackets with cocycles.
Connections with the construction of compatible brackets found in the theory
of integrable systems are also briefly discussed
Symmetry breaking for toral actions in simple mechanical systems
For simple mechanical systems, bifurcating branches of relative equilibria
with trivial symmetry from a given set of relative equilibria with toral
symmetry are found. Lyapunov stability conditions along these branches are
given.Comment: 25 page
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
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