On the Existence of Global Variational Principles

In studying physical phenomena one frequently encounters differential equations which arise from a variational principle, i.e. the equations are the Euler-Lagrangequations obtained from the fundamental (or action) integral of a problem in the calculus of variations. Because solutions to the Euler-Lagrange equations determine the possible extrema of the fundamental integral, the first step in the solution of a given problem in the calculus of variations is to obtain the appropriate Euler-Lagrange equations. This state of affairs suggests the so-called inverse problem, viz. does a given differential equation arise from a variational principle and, if so, what is the Lagrangian for that principle? In addition to being of intrinsic interest, this problem is relevant to the study of differential equations. For example, the existence of a variational principle frequently leads to first integrals (via Noether\u27s theorem) and solutions by quadrature, it guarantees the existence of a well developed canonical formalism (see, e.g. Dedecker [9], Goldschmidt and Sternburg [11], Hermann [14] and Rund [27]), and it has important implications regarding the existence and bifurcation of solutions (Vainberg [33], Berger [7], Rabinowitz [25]). The inverse problem also has applications in numerical analysis in view of the increasing popularity of the so-called direct methods such as the finite element method (see, e.g. Mitchell and Wait [22]). Finally, within the context of physical field theories there is considerable interest in this problem because of the belief that physically meaningful field equations should be EulerLagrange equations