时滞Lagrange系统的Lie对称性与守恒量研究

研究了位形间中含单时滞参数的非保守力学系统的Lie对称性和守恒量。首先，利用含时滞的动力学Hamilton原理，建立了含时滞的非保守系统的分段Lagrange运动方程；其次，利用微分方程容许Lie群理论，得到系统的Lie对称确定方程；然后，根据对称性与守恒量之间的关系，通过构造结构方程，得到含时滞的非保守系统的Lie定理；最后，给出了两个具体的算例说明了方法的应用。结果表明:时滞参数的存在使非保守系统的Lagrange方程呈现分段特性，相应的Lie对称性确定方程的个数应是自由度数目的2倍，这对生成元函数提出了更高的限制，同时，守恒量呈现依赖速度项的分段表达。

Lie Symmetries and Conserved Quantities of Lagrangian Systems With Time Delays

The Lie symmetries and conserved quantities of non-conservative mechanical systems with time delays in configuration space were studied. Firstly, the piecewise Lagrangian equations for non-conservative systems with time delays were established according to the Hamiltonian principle of dynamics with time delay. Secondly, the determining equations of the Lie symmetry were obtained by means of the permissible Lie group theory for differential equations. Then, according to the relationship between symmetries and conserved quantities, the Lie theorem of non-conservative systems with time delays was obtained through construction of structural equations. Finally, 2 examples were given to illustrate the application of the method. The results show that, the time delay makes the Lagrangian equations of non-conservative systems piecewise, and the number of determining equations for Lie symmetry is twice of the number of degrees of freedom, which imposes higher restrictions on the generator functions. Meanwhile, the conserved quantity is also in a piecewise expression depending on the velocity term.