辛体系下含对称破缺因素动力学系统的近似守恒律

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.