A new method for establishing motion equations in nonholonomic systems

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 of nonholonomic 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.