§1. I N T R O D U C T I O N A long-standing problem in mathematical physics, posed first in 1887 by H. Helmholtz, is the determination and classification of variational principles for systems of differential equations. This is the inverse problem of the calculus of variations. The last twenty years have witnessed a renewed interest in this inverse problem and substantial progress has been recorded with regard to each of the following specific aspects of the problem: (i) the derivation and analysis of the Helmholtz conditions as necessary and (locally) sufficient conditions for a differential operator to coincide with the Euler-Lagrange operator for some Lagrangian (ii) the characterization of the obstructions to the existence of global variational prin- ciples for differential operators defined on manifolds (iii) the invariant inverse problem for differential operators with symmetry and (iv) the variational multiplier problem wherein variational principles are found, not for the given differential operator, but rather for the differential equations determined by that operator. In this paper we use the variational bicomplex and the theory of exterior differential sys- tems to study the variational multiplier problem (iv) for systems of ordinary differential equations. To review some of the recent developments with regard to problems (i) - (iv) and to describe the results of the present paper, let us consider the simplest problem in the calculus of variations. Throughout this paper we shall let x denote a single independent variable, u% be the dependent variables for i = 1, 2, . . . , m and let ii1 and ul be the first and second derivatives of the dependent variables. Derivatives of order k will be denoted by u\. For many of the examples in this paper, we take m — 2 and write u — u1 and v = u2 for the dependent variables. Now, given the Lagrangian L = L(x,ui,ui), (1.1) Received by the editor October 29, 1990 and revised June 11, 1991. Research supported by NSF Grant DMS 87-0833 and grants from the University of Toledo and Utah State University Offices of Research. 1

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