On Some Developments in Direct Methods of the Calculus of Variations

One of the significant events in mathematical physics, in this century, is the introduction and further development of the so-called direct methods which were first applied by Rayleigh and Ritz to possibly extremum but at least stationary variational problems; they have been extended by Galerkin to problems which are not even stationary but involve only variations in the sense of the principle of virtual work. It is shown in this paper how, in the course of a further development of direct methods, the question of a proper choice of coordinate functions and of a proof of convergence of the method in the case of nonextremum and nonstationary variational functionals have been solved. Since an application of direct methods depends largely on the availability of basic functionals preferably with at least the property of stationarity, it is shown how such functionals can be obtained by switching from the conventional energy space to more abstract spaces involving adjoint problems or variations of operators rather than functions. Also, the question of an application of direct methods to initial value problems has been considered, as well as a modification of Galerkin’s equations which allows one to avoid cumbersome boundary conditions. To sum up, one can say: the paper shows how recent research has made direct methods much more general and more broadly applicable than was the case at the time of their introduction to mathematical physics at the beginning of this century.