Lagrange multipliers in infinite dimensional spaces, examples of applications

The Lagrange multipliers method is used in mathematical analysis, in mechanics, in economics, and in several other fields, to deal with the search of the global maximum or minimum of a function, in the presence of a constraint. The usual technique, applied to the case of finitedimensional systems, transforms the constrained optimization problem into an unconstrained one, by means of the introduction of one or more multipliers and of a suitable Lagrangian function, to be optimized. In mechanics, several optimization problems can be applied to infinitedimensional systems. Lagrange multipliers method can be applied also to these cases.