Mechanical Systems II. Method of Lagrange Multipliers

Abstract. So far our approach to classical mechanics was limited to finding a critical point of a certain functional. In this lecture I shall present a first-order necessary condition for an extremum problem with additional equality constraints. This is so called a method of Lagrange multipliers. 1 Lagrange Conditions We shall deal with a first-order necessary condition for an extremum problem with equality constraints. The main result presented in this chapter is the La-grange Multiplier Theorem. To better understand the idea of the underlying theorem we shall limit ourself to functions of two variables and one equal-ity constraint. I shall not focus on exact properties of the functions used below, I will merely assume that all considered functions are continuously differentiable. We shall regard a problem of finding the minimum x ∗ = (x∗1, x 2) of a function f: R2 → R subject to a constraint equation h(x) = 0, h: R2 → R. (1) It will be vital for further analysis to introduce a notion of a level set. The level set through a point x ∗ is the set {x ∈ R2: h(x) = h(x∗)}. Let h(x∗) = 0 (i.e. x ∗ satisfies the constraint equation) and the differential dh(x∗) = ∂h(x1, x2