An exact sequence resolving the Euler operator of the calculus of variations for partial differential polynomials in several dependent and independent variables is described. This resolution provides a solution to the ‘Inverse problem of the calculus of variations’ for systems of polynomial partial equations.

That problem consists of characterizing those systems of partial differential equations which arise as the Euler-Lagrange equations of some variational principle. It can be embedded in the more general problem of finding a resolution of the Euler operator. In (3), hereafter referred to as I, a solution of this problem was given for the case of one independent and one dependent variable. Here we generalize this resolution to several independent and dependent variables simultaneously. The methods employed are similar in spirit to the algebraic techniques associated with the Gelfand-Dikii transform in I, although are considerably complicated by the appearance of several variables. In particular, a simple algebraic proof of the local exactness of a complex considered by Takens(5), Vinogradov(6), Anderson and Duchamp(1), and others appears as part of the resolution considered here.