一般完整系统Mei对称性的逆问题

动力学逆问题是星际航行学、火箭动力学、规划运动学理论的基本问题. Mei对称性是力学系统的动力学函数在群的无限小变换下仍然满足系统原来的运动微分方程的一种新的不变性. 本文研究广义坐标下一般完整系统的Mei对称性以及与Mei对称性相关的动力学逆问题. 首先, 给出系统动力学正问题的提法和解法. 引入时间和广义坐标的无限小单参数变换群, 得到无限小生成元向量及其一次扩展. 讨论由n个广义坐标确定的一般完整力学系统的运动微分方程, 将其Lagrange函数和非势广义力作无限小变换, 给出系统运动微分方程的Mei对称性定义, 在忽略无限小变换的高阶小量的情况下得到Mei对称性的确定方程, 借助规范函数满足的结构方程导出系统Mei对称性导致的Noether守恒量. 其次, 研究系统Mei对称性的逆问题. Mei对称性的逆问题的提法是: 由已知守恒量来求相应的Mei对称性. 采取的方法是将已知积分当作由Mei对称性导致的Noether守恒量, 由Noether逆定理得到无限小变换的生成元, 再由确定方程来判断所得生成元是否为Mei对称性的. 然后, 讨论生成元变化对各种对称性的影响. 结果表明, 生成元变化对Noether和Lie对称性没有影响, 对Mei 对称性有影响, 但在调整规范函数时, 若满足一定条件, 生成元变化对Mei对称性也可以没有影响. 最后, 举例说明结果的应用.

Inverse problem of Mei symmetry for a general holonomic system

Inverse problems in dynamics are the basic problems in astronautics, rocket dynamics, and motion planning theory, etc. Mei symmetry is a kind of new symmetry where the dynamical function in differential equations of motion still satisfies the equation's primary form under infinitesimal transformations of the group. Mei symmetry and its inverse problem of dynamics for a general holonomic system in generalized coordinates are studied. Firstly, the direct problem of dynamics of the system is proposed and solved. Introducing a one-parameter infinitesimal transformation group with respect to time and coordinates, the infinitesimal generator vector and its first prolonged vector are obtained. Based on the discussion of the differential equations of motion for a general holonomic system determined by n generalized coordinates, their Lagrangian and non-potential generalized forces are made to have an infinitesimal transformation, the definition of Mei symmetry about differential equation of motion for the system is then provided. Ignoring the high-order terms in the infinitesimal transformation, the determining equation of Mei symmetry is given. With the aid of a structure equation which the gauge function satisfies, the system's corresponding conserved quantities are derived. Secondly, the inverse problem for the Mei symmetry of the system is studied. The formulation of the inverse problem of Mei symmetry is that we use the known conserved quantity to seek the corresponding Mei symmetry. The method is: considering a given integral as a Noether conserved quantity obtained by Mei symmetry, the generators of the infinitesimal transformations can be obtained by the inverse Noether theorem. Then the question whether the obtained generators are Mei symmetrical or not is verified by the determining equation, and the effect of generators' changes on the symmetries is discussed. It has been shown from the studies that the changes of the generators have no effect on the Noether and Lie symmetries, but have effects on the Mei symmetry. However, under certain conditions, while adjusting the gauge function, changes of generators can also have no effect on the Mei symmetry. In the end of the paper, an example for the system is provided to illustrate the application of the result.