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.