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The geometric structure of complex fluids

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Higher order Lagrange-Poincar, and Hamilton-Poincar, reductions

Tudor Ratiu, François Gay-Balmaz

Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higher-order framework for Lagrangian and Hamiltonian reduction by symmetry in geometric mechanics. In particular, we obtain the r ...

2011

On a class of mean field solutions of the Monge problem for perfect and self-interacting systems

Philippe Choquard

The Monge problem [23], [27], as reformulated by Kantorovich [19], [20] is that of the transportation, at a minimum "cost", of a given mass distribu- tion from an initial to a final position during a given time interval. It is an optimal transport problem ...

EPFL2010

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

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

On A Class Of Mean Field Solutions Of The Monge Problem For Perfect And Self-Interacting Systems

Philippe Choquard

The Monge problem (Monge 1781; Taton 1951), as reformulated by Kantorovich (2006a, 2006b) is that of the transportation at a minimum "cost" of a given mass distribution from an initial to a final position during a given time interval. It is an optimal tran ...

2010

Introduction to Geometric Mechanics and AVI

Yves Weinand, Tudor Ratiu, François Marie Alain Demoures

The goal of this short presentation is to introduce Geometric Mechanics as well as Asynchronous Variational Integrators (AVI). The geometric point of view in mechanics combined with solid analysis has been a phenomenal success in linking various diverse ar ...

2010

Hamiltonian approach to hybrid plasma models

Cesare Tronci

The Hamiltonian structures of several hybrid kinetic-fluid models are identified explicitly, upon considering collisionless Vlasov dynamics for the hot particles interacting with a bulk fluid. After presenting different pressure-coupling schemes for an ord ...

2010

Infinite dimensional geodesic flows and the universal Teichmüller space

François Gay-Balmaz

The subject of this thesis lies in the intersection of differential geometry and functional analysis, a domain usually called global analysis. A central object in this work is the group Ds(M) of all orientation preserving diffeomorphisms of a compact manif ...

EPFL2009

On Singular Poisson Sternberg Spaces

We obtain a theory of stratified Sternberg spaces thereby extending the theory of cotangent bundle reduction for free actions to the singular case where the action on the base manifold consists of only one orbit type. We find that the symplectic reduced sp ...

2009

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