Mechanical Systems With Nonideal Constraints: Explicit Equations Without the Use of Generalized Inverses

When constraints are applied to mechanical systems, additional forces of constraint are produced that guarantee their satisfaction. The development of the equations of motion for constrained mechanical systems has been pursued by numerous scientists and mathematicians, like Appell @1#, Beghin @2#, Chetaev @3#, Dirac @4#, Gauss @5#, Gibbs @6#, and Hamel @7#. All these investigators have used as their starting point the D’Alembert-Lagrange Principle. This principle, which was enunciated first by Lagrange in his Mechanique Analytique, @8#, can be presumed as being, at the present time, at the core of classical analytical dynamics. D’Alembert’s principle makes an assumption regarding the nature of constraint forces in mechanical systems, and this assumption seems to work well in many practical situations. It states that the total work done by the forces of constraint under virtual displacements is always zero. In 1992 Udwadia and Kalaba @9# obtained a simple, explicit set of equations of motion, suited for general mechanical systems, with holonomic and/or nonholonomic constraints. Though their equations encompass time dependent constraints that are ~1! not necessarily independent, and ~2! nonlinear in the generalized velocities, their equations are valid only when D’Alembert’s principle is observed by the constraint forces. However, in many situations in nature, the forces of constraint in mechanical systems do not satisfy D’Alembert’s principle. As stated in Pars’s A Treatise on Analytical Dynamics @10#, ‘‘There are in fact systems for which the principle enunciated @D’Alembert’s principle# . . . does not hold. But such systems will not be considered in this book.’’ Such systems have been considered to lie beyond the scope of Lagrangian mechanics. Recently, Udwadia and Kalaba @11,12# have developed general, explicit equations of motion for constrained mechanical systems that may or may not satisfy D’Alembert’s principle. The statement of their