Relationship between the Udwadia—Kalaba equations and the generalized Lagrange and Maggi’s equations

Abstract

In their paper “A new perspective on constrained motion” F. E. Udwadia and R. E. Kalaba offered a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. The obtained equations contain all the generalized coordinates of the mechanical system in question, and at the same time they do not contain the constraint reaction forces. This is an undoubted advantage of the equations presented, so the authors assume that “the equations of motion obtained in this paper appear to be the simplest and most comprehensive so far discovered”. To write these equations the authors apply a rather specific transformation proposed by Moore already in 1920 and developed by Penrose in 1955. In Russian literature it is said that in this case a pseudoinverse matrix is used. The present paper reveals that the equations obtained by those authors can be naturally derived from the generalized Lagrange and Maggi’s equations or when using a contravariant form of the equations of motion of a mechanical system subject to linear nonholonomic second-order constraints. It is noted that a similar technique for eliminating the reaction forces from differential equations is usually used in practical studying of motion of mechanical systems that are subject to holonomic and classical nonholonomic first-order constraints. As a result, we obtain the equations of motion containing only the generalized coordinates of a mechanical system, what corresponds to the equations of Udwadia—Kalaba form. Refs 7.