微扰力学系统近似守恒量与近似解的研究

If some minor factors ignored in the study of mechanics systems are considered, some perturbed terms will appear in motion differential equations of mechanics systems. Such mechanics systems containing perturbed terms are called perturbed mechanics systems. In spite of the smallness of the perturbed terms, their influences on the mechanics systems can be great. They can break some exact symmetries of the systems, and make the form of some exact conserved quantities change or even disappear, the exact solutions incorrect, and the stability of mechanics systems greatly influenced. Since the perturbed mechanics systems are much closer to the real mechanics systems, the findings has broader application.In the project, perturbed mechanics systems will be studied in the following points. 1. Based on the characteristic of the approximate conserved quantities, the recursion relations between the first order approximate conserved quantities to exact conserved quantities and the higher-order approximate conserved quantities to lower-order approximate conserved quantities are established, and a new method to obtain approximate conserved quantities is put forward; 2. The approximate Mei symmetry theory is established based on Mei symmetry theory so as to improve the theoretical study of the approximate symmetries and approximate conserved quantities and explain the reasons of broken symmetries and production of the approximate conserved quantities; 3. A study is made of the approximate solution of the perturbed mechanics systems by using Adomian decomposition method, and the stability of mechanics systems in this project is discussed; 4.The results of the study can be spread and applied in the study of such fields as engineering technology, celestial mechanics and atomic and molecular physics.

若将研究力学系统时忽略的某些次要因素重新加以考虑，则力学系统的运动微分方程中就会出现一些微扰项，存在微扰项的力学系统可称为微扰力学系统。虽然微扰项是小的，但其对力学系统的影响可能是大的，它会使力学系统的对称性遭到破损，一些精确守恒量的形式发生变化或消失，精确解不再成立，稳定性受到影响。微扰力学系统更接近实际力学系统，其研究成果更有推广价值。本项目对微扰力学系统开展以下研究：1）根据近似守恒量的性质，建立一阶近似守恒量与精确守恒量、高阶近似守恒量与低阶守恒量的递推关系，提出一种求近似守恒量的新方法；2）在Mei对称性理论的基础上，建立近似Mei对称性理论，以完善近似对称性与近似守恒量的研究理论，并解释对称性破损和产生近似守恒量的原因；3）运用Adomian分解法研究微扰力学系统的近似解，并讨论系统的稳定性。4）将研究成果推广应用于工程技术、天体力学、原子与分子物理等领域的研究。

许多实际力学系统的某些参数常常会随着位移、速度和时间发生微小的变化，即力学系统受到微扰作用，这样的力学系统称为微扰力学系统，此类系统的近似对称性和近似守恒量研究对于研究力学系统的特性至关重要。目前，研究微扰力学系统近似对称性与近似守恒量已有两种近似对称性法: 近似Lie对称性法和近似Noether对称性法，引进近似的群无限小变换，微分方程在此变换下近似保持不变则为近似Lie对称性；哈密顿作用量在此变换下近似保持不变则为近似Noether对称性，所得的守恒量为近似守恒量。这两种近似对称性法是在Lie对称性法和Noether对称性法的基础上发展起来的。因此，我们在Mei对称性理论的基础上建立了与Mei对称性相对应的近似Mei对称性理论，即引进近似的群无限小变换，力学系统的动力学函数在此变换下近似满足运动方程为近似Mei对称性，并研究了三种近似对称性之间的关系，进一步完善了近似对称性理论。三种近似对称性方法，都是先研究微扰力学系统的近似对称性，再研究相应的近似守恒量，研究的是微扰力学系统的正问题。我们从近似守恒量性质出发，提出了求近似守恒量的两种新方法：直接积分法和泊松括号法，并建立了高阶近似守恒量与低阶近似守恒量间的递推关系。我们提出的这两种方法，都是先研究微扰力学系统的近似守恒量，再研究与近似守恒量相对应的近似对称性，研究的是微扰力学系统的逆问题。我们还用Adomian分解法研究了一类分子分母都含非线性项的强非线性微分方程的近似解，所得近似解的解曲线与直接用Mathematica软件作出的解曲线以及与同伦渐近法所得解曲线进行比较，以说明Adomian分解法的优越性。本项目还对若干非线性系统的新的对称约化解、新的贝克隆变换、可积性和非局域对称等问题作了研究。