有多余坐标完整系统的自由运动

对于完整力学系统，若选取的参数不是完全独立的，则称为有多余坐标的完整系统. 由于完整力学系统的第二类Lagrange 方程中没有约束力，故为研究完整力学系统的约束力，需采用有多余坐标的带乘子的Lagrange方程或第一类Lagrange 方程. 一些动力学问题要求约束力不能为零，而另一些问题要求约束力很小. 如果约束力为零，则称为系统的自由运动问题. 本文提出并研究了有多余坐标完整系统的自由运动问题. 为研究系统的自由运动，首先，由d'Alembert-Lagrange 原理, 利用Lagrange 乘子法建立有多余坐标完整系统的运动微分方程；其次，由多余坐标完整系统的运动方程和约束方程建立乘子满足的代数方程并得到约束力的表达式；最后，由约束系统自由运动的定义，令所有乘子为零，得到系统实现自由运动的条件. 这些条件的个数等于约束方程的个数，它们依赖于系统的动能、广义力和约束方程，给出其中任意两个条件，均可以得到实现自由运动时对另一个条件的限制. 即当给定动能和约束方程，这些条件会给出实现自由运动时广义力之间的关系. 当给定动能和广义力，这些条件会给出实现自由运动时对约束方程的限制. 当给定广义力和约束方程，这些条件会给出实现自由运动时对动能的限制. 文末，举例并说明方法和结果的应用. 

FREE MOTION OF HOLONOMIC SYSTEM WITH REDUNDANT COORDINATES

If the parameters are not completely independent for holonomic systems, it is called holonomic systems with redundant coordinates. In order to study the forces of constraints for holonomic systems, we use the Lagrange equations with multiplicators of redundant coordinates or the first kind of Lagrange equations. Because there are no forces of constraints in the second kind of Lagrange equations. In some mechanical problems, the forces of constraints should not be equal to zero. In other conditions, the forces of constraints are very tiny. However, if the forces of constraints are all equal to zero, we called the free motion of constraints mechanical systems. This paper presents the free motion of holonomic system with redundant coordinates. At first, the differential equations of motion of the system are established according to d’Alembert-Lagrange principle. Secondly, the form of forces of constraints is determined by using the equations of constraints and the equations of motion. Finally, the condition under which the system has a free motion is obtained. The number of this conditions is equal to the constraints equation’s, its depend on the kinetic energy, generalized forces and constraints equations. If the two arbitrary conditions are given, the third one should be obtained when the system becomes free motion. At the end, some examples are given to illustrate the application of the methods and results.