Lagrangian mechanics, Gauss’ principle, quadratic programming, and generalized inverses: new equations for non-holonomically constrained discrete mechanical systems

In this paper we formulate Lagrangian mechanics as a constrained quadratic minimization problem. This quadratic minimization problem is then solved using the theory of generalized inverses of matrices thereby obtaining the explicit equations of motion of constrained, discrete mechanical systems. The approach extends the boundaries of Lagrangian mechanics in that we provide a general formulation for describing the constrained motion of such systems without either the use of Lagrange multipliers or the use of quasi-coordinates. An important feature of the approach is that we do not require prior knowledge of the specific set of constraints to accomplish this formulation. This makes the equations presented here more generally useful, and perhaps more aesthetic, than the Gibbs-Appell equations which require a felicitous choice of problem-specific quasi-coordinates. The new equations of motion presented here are applicable to both the holonomic and nonholonomic constraints that Lagrangian mechanics deals with. They are obtained in terms of the usual generalized coordinates used to describe the constrained system. Furthermore, they can be integrated by any of the currently available numerical integration methods, thus yielding analytical and/or computational descriptions of the motions of constrained mechanical systems.