对称性破缺的约化理论及其在非线性系统中的应用
Reduction theory for symmetry breaking with applications to nematic systems
摘要：
我们制定Euler-Poincaré和Lagrange-Poincaré方程的对称性破缺系统。我们专门的一般理论，提出明确的运动方程的双折射系统，从单一的双折射分子的双轴液晶。几何构造适用于由无符号单位向量（导向器）或对称矩阵（对齐张量）组成的序参量空间。在汉密尔顿方面，我们在Lie-Poisson和Hamilton-Poincaré公式中提供了相应的Poisson括号。单轴向列的螺旋度不变量的显式形式，连同一个完整的类不变量（Casimirs）的二维不可压缩流。
英文摘要：
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.
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