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Approximate conservation laws for dynamic systems with symmetry breaking in symplectic framework

DOI：10.7511/jslx20230831001

 作者 单位 E-mail 胡伟鹏 西安理工大学 土木建筑工程学院, 西安 710048 wphu@nwpu.edu.cn 林志华 香港城市大学 建筑学及土木工程学系, 香港 999077 邓子辰 西北工业大学 力学与土木建筑学院, 西安 710072

自冯康先生创立Hamilton系统辛几何算法以来，诸如辛结构和能量守恒等守恒律逐渐成为动力学系统数值分析方法有效性的检验标准之一。然而，诸如阻尼耗散、外部激励与控制和变参数等对称破缺因素是实际力学系统本质特征，影响着系统的对称性与守恒量。因此，本文在辛体系下讨论含有对称破缺因素的动力学系统的近似守恒律。针对有限维随机激励Hamilton系统，讨论其辛结构；针对无限维非保守动力学系统、无限维变参数动力学系统、Hamilton函数时空依赖的无限维动力学系统和无限维随机激励动力学系统，重点讨论了对称破缺因素对系统局部动量耗散的影响。上述结果为含有对称破缺因素的动力学系统的辛分析方法奠定数学基础。

Since the establishment of the symplectic geometric method for Hamiltonian systems by K.Feng, a globally recognized, prominent mathematician and scientist, the conservation laws including symplectic structures and energy conservation have become one of the effective verification criteria for numerical approaches of dynamic systems.However, some intrinsic system characteristics including damping dissipation, external excitation and control, variable coefficients, etc., that cause symmetry breaking in practical dynamic systems affect the system symmetry and conservation laws.In this paper, the approximate conservation laws of dynamic systems considering various symmetry breaking factors are analyzed in detail.Based on the geometric symmetry theory, the symplectic structure for finite-dimensional stochastic dynamic systems is obtained.Further, for infinite-dimensional non-conservative dynamic systems with various coefficients, time-space dependent Hamilton functions, and stochastic dynamic systems, the effects of symmetry breaking factors on local energy dissipation are investigated.The result established here may form the mathematical basis for symplectic analysis of dynamic systems with broken symmetry.