§2. T H E VARIATIONAL B I C O M P L E X FO R ORDINAR Y D I F F E R E N T I A L EQUATION S The variational bicomplex was first introduced to provide a differential geometric frame- work for the calculus of variations. It incorporates naturally the Euler-Lagrange operator and the Helmholtz conditions and it provides an effective means for solving the global in- verse problem of the calculus of variations. Our use of the variational bicomplex to study the variational multiplier problem is a new application. We begin by quickly outlining the basic construction of the variational bicomplex, adapted to the particular needs of this paper. For a full development of this subject, see Anderson [2]. Since all our considerations are of a purely local nature, we need only construct the variational bicomplex for the trivial bundle E: R x R m R. Coordinates on E are (x,ul) —• x. Let 7TQ : Jk(E) - R be the k-th. jet bundle of E. A point in Jk(E) is an equivalence class of local sections of E\ two local sections si, 52 : (a, b) —• E, defined on an open interval (a, 6), are equivalent at a point XQ G (a, b) if s\ and S2 and their derivatives with respect to x agree at XQ to order k. The equivalence class of a section s at XQ is denoted by jk(s)(x0). There are natural projections Trf: Jk(E)-*j\E) for k I. The inverse limit of the system {Jk(E),7vk} defines the infinite order jet bundle 7r0°°: J ° ° ( £ ) - + R . The coordinates of a point J°°(S)(XQ) are j°°0)(:ro) = 0 o , u\ u\, u 2 , . . . , uj.,... ), where .• drsi{x)\ Ur=Z J^r 12
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