What is the General Form of the Explicit Equations of Motion for Constrained Mechanical Systems ?

Since its inception more than 200 years ago, analytical mechanics has been continually drawn to the determination of the equations of motion for constrained mechanical systems. Following the fundamental work of Lagrange @1# who bequeathed to us the so-called Lagrange multipliers in the process of determining these equations, numerous scientists and mathematicians have attempted this central problem of analytical dynamics. A comprehensive reference list would run into several hundreds; hence we shall provide here, by way of a thumbnail historical review of the subject, only some of the significant milestones and discoveries. In 1829, Gauss @2# introduced a general principle for handling constrained motion, which is commonly referred to today as Gauss’s Principle; Gibbs @3# and Appell @4# independently obtained the so-called Gibbs-Appell equations of motion using the concept of ~felicitously chosen! quasi-coordinates; Poincare @5#, using group theoretic methods, generalized Lagrange’s equations to include general quasi-coordinates; and Dirac @6#, in a series of papers provided an algorithm to give the Lagrange multipliers for constrained, singular Hamiltonian systems. Udwadia and Kalaba @7# gave the explicit equations of motion for constrained mechanical systems using generalized inverses of matrices, a concept that was independently discovered by Moore @8# and Penrose @9#. The use of this powerful concept, which was further developed from the late 1950s to the 1980s, allows the generalized-inverse equations ~Udwadia and Kalaba @7#! to go beyond, in a sense, those provided earlier; for, they are valid for sets of constraints that could be nonlinear in the generalized velocities, and that could be functionally dependent. Thus the problem of obtaining the equations of motion for constrained mechanical systems has a history that is indeed as long as that of analytical dynamics itself. Yet, all these efforts have been solely targeted towards obtaining the equations of motion for holonomically and nonholonomically constrained systems that all obey D’Alembert’s principle of virtual work at each instant of time. This principle, though introduced by D’Alembert, was precisely stated for the first time by Lagrange. The principle in effect makes an assumption about the nature of the forces of constraint that act on a mechanical system.