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Structure-Preserving Reduced Basis Methods For Poisson Systems

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Symmetry Reduced Dynamics of Charged Molecular Strands

Tudor Ratiu, François Gay-Balmaz

The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonloca ...

Springer-Verlag2010

The Momentum Map, Symplectic Reduction and an Introduction to Brownian Motion

The underlying goal of this Master's thesis is of laying down, in so far as possible, the foundations for later work in Geometric Stochastic Mechanics. The first part is a presentation of symplectic reduction, going through the momentum map and culminating ...

2010

Reduction theory for symmetry breaking with applications to nematic systems

François Gay-Balmaz, Cesare Tronci

We formulate Euler-Poincare and Lagrange-Poincare equations for systems with broken symmetry. We specialize the general theory to present explicit equations of motion for nematic systems, ranging from single nematic molecules to biaxial liquid crystals. Th ...

2010

The Momentum Map In Poisson Geometry

Tudor Ratiu, Juan-Pablo Ortega

Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction. ...

2009

The geometric structure of complex fluids

Tudor Ratiu, François Gay-Balmaz

This paper develops the theory of affine Euler-Poincare and affine Lie-Poisson reductions and applies these processes to various examples of complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids, spin glasses, microf ...

Elsevier2009

Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids

Tudor Ratiu, François Gay-Balmaz

The Lagrangian and Hamiltonian structures for an ideal gauge-charged fluid are determined. Using a Kaluza-Klein point of view, the equations of motion are obtained by Lagrangian and Poisson reductions associated to the automorphism group of a principal bun ...

2008

Poisson reduction and the Hamiltonian structure of the Euler-Yang-Mills equations

Tudor Ratiu, François Gay-Balmaz

The problem treated here is to find the Hamiltonian structure for an ideal gauge-charged fluid. Using a Kaluza-Klein point of view, we obtain the non-canonical Poisson bracket and the motion equations by a Poisson reduction involving the automorphism group ...

American Mathematical Society, P.O. Box 6248 Ms. Phoebe Murdock, Providence, Ri 02940 Usa2008

Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrodinger Equation

Charles Stuart

In the first part of these notes, we deal with first order Hamiltonian systems in the form Ju'(t) = del H(u(t)) where the phase space X may be in infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian H is an element ...

Birkhäuser-Verlag, Springer2008

Affine Lie-Poisson reduction, Yang-Mills magnetohydrodynamics, and superfluids

Tudor Ratiu, François Gay-Balmaz

This paper develops the theory of affine Lie-Poisson reduction and applies this process to Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids. As a consequence of this approach, the associated Poisson brackets are obtained by reduction of ...

2008

Induced and coinduced Banach Lie-Poisson spaces and integrability

Tudor Ratiu

The Poisson induction and coinduction procedures are used to construct Banach Lie-Poisson spaces as well as related systems of integrals in involution. This general method applied to the Banach Lie-Poisson space of trace class operators leads to infinite H ...

2008