One considers the integral functional Ĥ(y), depending on the mapping of the domain Ω⊂R^m into R^m, on the set of mappings y, subjected to the incompressibility condition: det y=1. One computes its first and second variations. The obtained results are compared with the formulas arising from the formal application of the method of the undetermined Lagrange multipliers. One gives an application to problems of elasticity theory.