Reduction theory for symmetry breaking with applications to nematic systems |
| |
| Authors: | Fran?ois Gay-Balmaz Cesare Tronci |
| |
| Institution: | 1. Control and Dynamical Systems, California Institute of Technology, United States;2. Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Switzerland;1. Nansen Environmental and Remote Sensing Center, Thormøhlens gate 47, Bergen, Norway;2. Geophysical Institute, University of Bergen, Allégaten 70, Bergen, Norway;3. Bjerknes Center for Climate Research, Bergen, Norway;4. Department of Meteorology, Stockholm University, S-106 91 Stockholm, Sweden;5. Department of Oceanography, University of Cape Town, 7701 Rondebosch, South Africa;6. Nansen-Tutu Centre for Marine Environmental Research, 7701 Rondebosch, South Africa;1. Department of Mathematics, University of Hawaii at Manoa, United States of America;2. CNRS - LMD, Ecole Normale Supérieure, France;1. Department of Mathematics, Imperial College London, UK;2. Met Office, Exeter, UK;3. Department of Computing, Imperial College London, UK |
| |
| 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. |
| |
| Keywords: | |
| 本文献已被 ScienceDirect 等数据库收录！ |
|
