奇异系统对称性及其摄动理论研究

由于物理系统大多是奇异系统，其运动实际上总是受到约束的限制，因此，恰当地对约束进行分析和处理至关重要。本项目采用分析力学的方法，对约束Hamilton系统及场论中奇异系统的变分问题、对称性与守恒量、对称性的摄动与绝热不变量进行研究，所得结果用于进一步分析该系统的Dirac约束、规范生成元及Dirac猜想等。考虑到分数阶模型往往比整数阶模型更贴合实际，本项目将基于分数阶Dirac理论，研究分数阶约束Hamilton系统和场论中奇异系统的整体对称性（如Noether对称性、Lie对称性、Mei对称性），给出相应的守恒量；研究其定域对称性，给出Noether第二定理；研究该系统对称性的摄动与绝热不变量。时间尺度为解决复杂动力学行为提供了新途径，本项目将基于时间尺度上的Dirac理论，研究时间尺度上约束Hamilton系统和场论中奇异系统的整体对称性、定域对称性及对称性的摄动与绝热不变量。

Physical systems are mostly singular systems. Because the movement of physical systems is always restricted by constraints, it is very important to analyze and process those constraints properly. Methods developed in analytical mechanics are used in this project to study the variational problems, symmetries and conserved quantities, as well as symmetrical perturbation and adiabatic invariants for the constrained Hamiltonian systems and the singular systems in field theory. And the obtained results are used for the further analysis of Dirac constraints, gauge generators and Dirac conjecture of the systems. The fractional order model is often more practical than the integer order model, consequently, based on the fractional Dirac theory, this project will investigate the fractional order constrained Hamiltonian systems and the singular systems in field theory, including the global symmetry (such as Noether symmetry, Lie symmetry and Mei symmetry), presenting the corresponding conserved quantity; the local symmetry, giving the second Noether theorem; and the symmetrical perturbation and adiabatic invariants. Time scale provides a new way to solve the complicated dynamics behaviors, therefore, based on the Dirac theory on time scales, the global symmetry, the local symmetry and the symmetrical perturbation and adiabatic invariants will be studied for the constrained Hamiltonian systems and the singular systems in field theory on time scales.