Contribution to Book
M. Gotay, J. Marsden, V. Moncrief
The variational bicomplex was first introduced in the mid 1970's as a means of studying the inverse problem of the calculus of variations. This is the problem of characterizing those differential equations which are the Euler-Lagrange equations for a classical, unconstrained variational problem. Since then, the variational bicomplex has emerged as an effective means for studying other formal, differential-geometric aspects of the calculus of variations. Moreover, it has been shown that the basic variational bicomplex constructed to solve the inverse problem can be modified in various ways and that the cohomology groups associated with these modified bicomplexes are relevant to many topics in geometry, mathematical physics and differential equations. The purpose of this paper is to review the general construction of the variational bicomplex, to describe some of its the basic properties, and to survey some recent results.
Introduction to the Variational Bicomplex, in Mathematical Aspects of Classical Field Theory (ed. byM. Gotay, J.Marsden, V.Moncrief), Comptem- porary Mathematics Vol 132, 1992.