Symmetry, geometry and numerics of conservative dynamics

保守动力学的对称性、几何和数值

• 批准号: 105716-2011
• 负责人: Patrick, George
• 金额: $ 0.73万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569104
• 关键词: Symmetry; geometry; numerics; conservative; dynamics

Conservative systems model physical phenomena where friction is not dominant. In the models, which are typically ordinary differential equations of moderate dimension, friction is absent, multiple time scales are present, and the long-time dynamics is complicated. Modern applications include underwater vehicle dynamics, molecular dynamics and spectra, fluids and plasmas, physical field theories, and even image analysis. Included are physical systems which roll without slipping, such as ordinary road vehicles. The analysis of these systems depends on fundamental differential-geometric structures that define their differential equations, and many of the interesting systems have symmetry, in the form of an action of a Lie group. Computer simulations are critical for experimentation and validation. In my research, Mathematics, Physics, and Computer Science, are combined to obtain new results and understandings about the structures and dynamics. Geometrically exact discrete versions of these systems, interesting in themselves, provide new numerical algorithms and high fidelity long time simulations. Key recent progress: (1) Precise tests for the stability of relative equilibria in the case of noncompact symmetry, based on the non-Hausdorff topology of the coadjoint orbit space of noncompact group actions. (2) The identification of structural gaps, where energy-momentum Liapunov functions fail. (3) Recovery of nonlinear stability using KAM theory, verifications of dissipation-induced instability in the gaps, and applications to the stability of relative equilibria of underwater vehicles. (4) A development of the structures of nonholonomic systems using variational methods. (5) New geometric discretizations of tangent bundles, and associated numerical analysis of the accuracy of the corresponding variational integrators. (6) A software system to automatically generate variational integrators.

保守系统模拟摩擦不占主导地位的物理现象。该模型是典型的中维常微分方程，不存在摩擦，存在多时间尺度，长时间动力学复杂。 现代应用包括水下航行器动力学，分子动力学和光谱，流体和等离子体，物理场理论，甚至图像分析。包括滚动而不打滑的物理系统，如普通道路车辆。 这些系统的分析依赖于定义其微分方程的基本微分几何结构，并且许多有趣的系统具有对称性，以李群作用的形式。计算机模拟对于实验和验证至关重要。在我的研究中，数学，物理学和计算机科学相结合，以获得新的结果和理解的结构和动力学。这些系统的几何精确的离散版本，有趣的是，提供了新的数值算法和高保真度的长时间模拟。 最近的主要进展：（1）基于非紧群作用的余伴随轨道空间的非豪斯多夫拓扑，对非紧对称情形下相对平衡点稳定性的精确检验。 (2)结构间隙的识别，其中能量动量李雅普诺夫函数失败。(3)利用KAM理论恢复非线性稳定性，验证间隙中耗散引起的不稳定性，并应用于水下航行器相对平衡的稳定性。(4)用变分法研究非完整系统的结构。(5)切丛的新的几何离散化，以及相应变分积分器精度的数值分析。(6)自动生成变分积分器的软件系统。