时间尺度上约束Hamilton系统的Noether对称性和守恒量

研究时间尺度上相空间中非保守奇异系统的Noether对称性和守恒量. 首先, 将奇异性导致的内在约束按外在非完整约束等效处理, 利用时间尺度上Δ导数下的Hamilton原理得到约束Hamilton系统的正则方程； 其次, 引进时间不变的特殊无限小变换, 得到系统Hamilton作用量在该变换下的Noether对称性的判据和定理； 最后, 举例说明该方法和结果的有效性. 结果表明, 时间尺度上约束Hamilton系统的正则方程结构属性依旧保持, 系统的奇异性使Noether对称性不再直接导致Noether类型的守恒量, 还需构造一定的规范函数使Noether对称性满足结构方程.

Noether Symmetries and Conserved Quantities of Constrained Hamiltonian Systems on Time Scales

The author studied the Noether symmetries and conserved quantities of non-conservative singular systems in phase space on time scales. Firstly, the internal constraints caused by singularity were treated as external non-holonomic constraints, and the canonical equations of constrained Hamiltonian systems were obtained by using Hamilton principle under Δ derivative on time scale. Secondly, the special infinitesimal transformation with time invariance was introduced, and the criterion and theorem of Noether symmetry were obtained according to the invariance of Hamilton action. Finally, the effectiveness of the method and results was  illustrated by an example. The results show that the structural properties of canonical equations of constrained Hamiltonian systems on time scale are still maintained, the singularity of the system makes the Noether symmetry no longer directly lead to the Noether-type conserved quantity, and it is necessary to construct a certain gauge function to make the Noether symmetry satisfy the structural equation.