This chapter is devoted in its entirety to the description of the so called "Lagrange undetermined multipliers method". This very elegant and powerful mathematical procedure was devised by Lagrange to deal with constrained systems without having to first eliminate the superfluous variables. The availability of such a method magnifies clearly the advantages of using the analytical, instead of the vectorial, approach to establish the equations of motion of most material systems. Indeed, the task of eliminating dependent variables of a system is often tedious, or even inextricable. Hence, it is soon realized that in the absence of the Lagrange's multipliers method, the efficiency of analytical mechanics would be severely limited to a fairly restricted class of problem. Here, the mathematical aspects of the method are described and then the physical contents are illustrated by discussing several examples of practical interest, which involve holonomic constraints, first scleronomic and then rheonomic. Since the latter case is also classically treated by using a change of reference frame, this provides us with a good opportunity to discuss these two points of view for handling mechanical systems subjected to prescribed motion.

4.1. Introduction

In the preceding two chapters, Lagrange's formalism was introduced within the restricted field of unconstrained systems; that means mechanical systems described by independent coordinates, or displacements, in the same number as the degrees of freedom of the system. In this chapter, Lagrange's formalism will be extended to the case of discrete systems which comply with constraint conditions and are described by...