Reduction theory for symmetry breaking with applications to nematic systems

Abstract

We formulate Euler-Poincaré and Lagrange-Poincaré 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. The geometric construction applies to order parameter spaces consisting of either unsigned unit vectors (directors) or symmetric matrices (alignment tensors). On the Hamiltonian side, we provide the corresponding Poisson brackets in both Lie-Poisson and Hamilton-Poincaré formulations. The explicit form of the helicity invariant for uniaxial nematics is also presented, together with a whole class of invariant quantities (Casimirs) for two dimensional incompressible flows.

40 pages. To appear in Physica D

References in corpus (2)

• Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions
• Geometric dynamics of optimization

Cited by in corpus (23)

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• Invariant higher-order variational problems
• Invariant higher-order variational problems II
• Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows
• Equivalent theories of liquid crystal dynamics
• Geometry and symmetry of quantum and classical-quantum variational principles
• Geometry of nonadiabatic quantum hydrodynamics
• Geometric dynamics on the automorphism group of principal bundles: geodesic flows, dual pairs and chromomorphism groups
• Euler-Poincaré approaches to nematodynamics
• Lagrangian reduction and wave mean flow interaction
• The helicity and vorticity of liquid crystal flows
• Dynamics of mixed quantum-classical spin systems
• Geometric theory of flexible and expandable tubes conveying fluid: equations, solutions and shock waves
• Hybrid models for complex fluids with multipolar interactions
• Lagrangian reductions and integrable systems in condensed matter
• From liquid crystal models to the guiding-center theory of magnetized plasmas
• Persistence of stationary motion under explicit symmetry breaking perturbation
• Lagrangian Reduction on Homogeneous Spaces with Advected Parameters
• Symplectic slice for subgroup actions
• Equivalent variational approaches to biaxial liquid crystal dynamics
• Holonomy and vortex structures in quantum hydrodynamics
• Dirac Reduction for Nonholonomic Mechanical Systems and Semidirect Products
• Optimal Control Problems with Symmetry Breaking Cost Functions