Natural Variational Principles on Riemannian Structures
Annals of Mathematics
In the calculus of variations, the Euler-Lagrange operator E(L) refers to the differential operator derived from the first variation of the fundamental (or action) integral. Solutions of the differential equations E(L) = 0 are thus the critical points of the fundamental integral. Conversely, given a differential operator, say T, it is natural to ask if T can be identified as the Euler-Lagrange operator derived from some Lagrangian L. This is the inverse problem to the calculus of variations. If the Lagrangian is invariant under the action of a Lie group G, then it is well known that the corresponding Euler-Lagrange equations are also invariant under G. Thus, the following equivariant inverse problem to the calculus of variations may be stated. If T is a differential operator which is (i) invariant under the action of a Lie group G; and (ii) the Euler-Lagrange operator of some Lagrangian, then when is T the Euler-Lagrange operator of a G-invariant Lagrangian?
Natural Variational Principles on Riemannian Structures, Annals ofMath- ematics 120 (1984), 329–370.