Elimination of Lagrangian multipliers (Q1087826)

In case of a mechanical system of rigid bodies (or more generally elastic ones) it is sometimes necessary to determine the reaction produced by the constraints. In this situation the most comfortable and used way to write the motion equations and to determine reactions is the Lagrangian multipliers method. This method can be applied in two ways: a) by using the d'Alembert principle with the Lagrangian multipliers (this way is generally used by engineers, as multipliers are usually the reaction due to the constraints); b) by using the Lagrangian equations with multipliers. The purpose of the present note is to show a ''natural'' and simple method of eliminating the Lagrangian multipliers and some comments on this method. It relies on the orthogonality between the virtual displacements, compatible with the constraints, and the reactions due to constraints. The paper is concerned with the way b), by which a more general description of the method can be achieved. For a proper choice of the equations expressing the constraints, the multipliers can be considered as the sought reactions.