本文分为两个部分，第一部分是对于脂质囊泡形状方程的首次积分的研究，第二部分是关于球面上活性向列相液晶拓扑缺陷的动力学。
在第一部分中，我们关心了轴对称脂质囊泡形状方程的积分问题。脂质囊泡的形状方程，是解释生物膜形状和力学特征的重要理论工具。对形状方程的解析求解，是生物物理学家和数学家关心的重要课题。根据郑伟谋和刘寄星的工作 (Phys. Rev. E 48 2856 (1993)), 在轴对称情况下，脂质囊泡形状方程可以从四阶偏微分方程变为二阶常微分方程。我们希望可以进一步找到方程的新的首次积分，把形状方程降为一阶方程，从而能更好地探讨其可积性。首先，我们把二阶轴对称脂质囊泡形状方程看作力学系统中的一个 Euler-Lagrange 方程。通过求解逆变分问题，我们得到了轴对称形状方程对应的 Lagrangian. 然后，我们利用 Noether 定理分析了轴对称形状方程 Lagrangian 的对称性，以及相对应的守恒量。我们发现，当形状方程的对称群存在特定限制时，方程存在首次积分。此时，形状方程等价为 Willmore 方程，方程首次积分对应的对称性为共形不变性。
文章的第二部分介绍了我们对球面上活性向列相液晶拓扑缺陷的研究。活性系统中出现的非平衡集体现象是软物质研究的重要课题，活性和拓扑约束的耦合往往会使系统呈现出丰富的物理现象。由于拓扑约束，束缚在球面上的向列相液晶系统中会形成总荷为 +2 的拓扑缺陷。当处于平衡态时，球面上向列相方向场的拓扑缺陷构型为四个 +1/2 的点缺陷，分别位于球内接正四面体的四个顶点，从而使系统的 Frank 自由能极小。对于活性向列相系统而言，每个液晶分子呈现出自驱动的定向运动，这会导致系统出现大尺度的活性流，从而使系统远离平衡态。 活性流的存在，使活性向列相中的缺陷表现出了复杂的动态轨迹。根据系统中所引入活性强度的不同，缺陷位置的轨迹可表现出不同的特征。对于弱活性，缺陷轨迹表现为良好的周期性轨道运动；对于强活性，缺陷轨迹逐渐过渡到混沌运动。通常，借助唯象粒子模型，可以部分理解活性液晶缺陷的奇特动力学行为。但是，目前为止，球面上活性向列相液晶缺陷的唯象理论尚未在微观层面得到完整解释。在本研究中，我们将尝试解决这个问题。在 Onsager 变分原理的框架下，通过把拓扑缺陷本身作为系统的自由度，我们推导出了球面上活性向列相液晶的缺陷运动的等效粒子理论，并建立了缺陷运动方程和活性液晶流体力学间的联系。通过对所得到运动方程的数值分析，我们证实了先前在实验中观测到的缺陷动力学特性可以被该理论解释，同时阐明了缺陷方向的时间演化对其动力学行为的重要影响。 我们的工作表明，Onsager 变分原理作为在非平衡系统中广泛适用的理论框架，对于我们理解活性液晶系统的性质起到了关键作用。Onsager 变分原理提供了一种极其直接有效的方式，使我们得以对活性液晶的流体力学进行合理的粗粒化，并且使得缺陷的迁移率可以在流体动力学水平上得到解释。

The thesis separates into two parts. The first part is the study of the first integrals of the axisymmetric shape equation of lipid membranes. The second one is about dynamics of active nematic defects on the surface of a sphere.
In the first part, we care about the integral of the axisymmetric shape equation of lipid membranes. The shape equation of lipid membranes is a crucial theoretical tool to explain the shape and mechanical characteristics of biological membranes. The analytical solution to the shape equation is an important subject of concern to biophysicists and mathematicians. The shape equation of lipid membranes is a fourth-order partial differential equation. Under the axisymmetric condition, this equation was transformed into a second-order ordinary differential equation (ODE) by Zheng and Liu (Phys. Rev. E 48 2856 (1993)). Here we try to further reduce this second-order ODE to a first-order ODE, so as to better explore its integrability. First, we invert the usual process of variational calculus, that is, we construct a Lagrangian for which the ODE is the corresponding Euler-Lagrange equation. Then, we seek symmetries of this Lagrangian according to the Noether theorem. Under a certain restriction on Lie groups of the shape equation, we find that the first integral only exists when the shape equation is identical to the Willmore equation, in which case the symmetry leading to the first integral is scale invariance.
The second part of the thesis presents our study about the topological defects of the active nematic defects on the surface of a sphere. The non-equilibrium collective phenomenon that appears in the active system is an important subject of the soft matter research. The coupling of activity and topological constraints often makes the system present a wealth of physical phenomena. A nematic liquid crystal confined to the surface of a sphere exhibits topological defects of total charge +2 due to the topological constraint. In equilibrium, the nematic field forms four +1/2 defects, located at the corners of a tetrahedron inscribed within the sphere, since this minimizes the Frank elastic energy. If additionally the individual nematogens exhibit self-driven directional motion, the resulting active system creates large-scale flow that drives it out of equilibrium. In particular, the defects now follow complex dynamic trajectories which, depending on the strength of the active forcing, can be periodic (for weak forcing) or chaotic (for strong forcing). Generally, with the help of phenomenological particle models, the novel dynamic behavior of active nematic defects can be partially understood. However, so far, the phenomenological theory of active nematic defects on the spherical surface has not been fully explained at the microscopic level. In this research, we will try to solve this problem. Based on the framework provided by Onsager's variational principle, we derive an effective particle theory for defects of the active nematic confined to the surface of a sphere, in which the topological defects are the degrees of freedom, whose exact equations of motion we subsequently connect with the hydrodynamics of the active nematic. Numerical solutions of these equations confirm previously observed characteristics of their dynamics and clarify the role played by the time dependence of their global rotation. We also show that Onsager's variational principle offers an exceptionally transparent way to derive these dynamical equations, and we explain the defect mobility at the hydrodynamics level.