一种建立非完整系统运动方程的新方法

A new method for establishing motion equations in nonholonomic systems

本文提出了一种不基于任何变分原理而建立非完整系统基本运动方程的新方法.利用动力学方程与广义坐标的选取以及非完整约束函数选取的无关性,结合矩阵的乘积规则,提出了用于表述力学系统协变性的双指标张量分析方法;将力学系统的首次积分等效为作用在系统上的非完整约束,说明了非完整系统的自洽性,进而根据非完整系统的运动方程在首次积分约束下的不变性、非完整约束反力在广义坐标和等效非完整约束函数组变换下的不变性,反推出了非完整约束反力所必须满足的形式,以此建立了非完整系统的基本运动方程.本文提出的方法完全基于非完整系统的自洽性与协变性,不仅没有使用任何先验的D’Alembert-Lagrange, Gauss, Jourdian或Hamilton变分原理,而且还为非完整系统Chetaev条件的成立提供一个合理的解释,并且从自洽性的角度说明了基于Hamilton原理所导出的Vakonomic力学不是非完整系统的合理模型.

In this paper, we present a new method of establishing the fundamental motion equations of nonholonomic systems without using any variational principles. According to the invariance of motion equations under the generalized coordinates transformation and the nonholonomic constraint function groups transformation, the double index tensor analysis method is proposed cooperating with the product rules of matrix. Then we let first integrals equivalent to the corresponding nonholonomic constraints imposed upon the system to explain the self consistency of nonholonomic systems, it is shown that the self consistency is the core of nonholonomic mechanics and can verify which model is appropriate for describing nonholonomic systems. Therefore, the form of nonholonomic constraint force is determined only by three qualities： the invariance of the motion equations under the constraint of first integrals, the invariance of nonholonomic constraint force under the generalized coordinates transformation and the nonholonomic constraint function groups transformation, to establish the fundamental motion equation of nonholonomic systems. Using the double index tensor analysis method, the three qualities and the Moore-Penrose generalized inverse of matrix theory, the equation of motion ofnonholonomic systems are derived. Some classical equations in nonholonomic systems, such as the Routh equation, the Nielsen equation and the Chaplygin equation are also derived to prove our method. The method is totally based on the self consistency of nonholonomic systems and the covariation of motion equations, these two characters are naturally derived from mathematical and mechanical requirements of nonholonomic systems. Not only does not use any transcendental variational principles, such as D＇Alembert-Lagrange principle, Gauss principle or Jourdian principle, but also illustrate that the validity of the three principles are attributed to the two characters. Further more, this method even provides a reasonable explanation for Chetaev conditions in nonholonomic systems, and shows that the vakonomic mechanics derived by Hamilton＇s principle is not an appropriate model for nonholonomic systems.