The Variational Bicomplex

Wednesday, January 29, 2014 - 1:30pm - 2:30pm
Lind 305
Irina Kogan (North Carolina State University)
Introduction of the variational bicomplex can be motivated by drawing
an analogy with vector calculus. It is well known that one can
reformulate vector calculus in terms of differential forms, and use
the exterior derivative to represent gradient, divergence, and curl
operators. Thus vector calculus can be efficiently expressed by the de
Rham complex. Similarly to vector calculus, many important aspects of
variational calculus, such as Euler-Lagrange operator, Helmholtz
operator, or Noether correspondence, can be formulated in terms of
complexes of differential forms. This leads to notion of the
variational bicomplex (and related notions of the variational complex,
the variational spectral sequence). These constructions originated
with the work of Dedecker (1957) and a large body of literature has
appeared afterwards. The purpose of my talk is to define the
variational bicomplex, explain how it encodes various aspects of
variational calculus and what role is played by its cohomology.