力学分析中的对称性与守恒律分析方法研究进展

This paper reviews the research progress on symmetry and conservation laws in mechanical analysis. It begins by introducing Lie group symmetries in continuous systems, including the symmetries of differential equations, partial differential equations, functionals, and approximate Lie symmetries of perturbed differential equations, with practical applications demonstrated through examples. The paper then explores symmetries and conservation laws in discrete systems, focusing on the dynamics equations, Noether symmetries, Lie symmetries, and Mei symmetries, with explanations supported by specific application examples. Finally, it reviews symmetries and conservation laws in stochastic systems, discussing the symmetries of Ito and Stratonovich stochastic differential equations, particularly in the statistical sense. The aim of this paper is to provide theoretical references for subsequent research and to advance the development of related fields.