Divergence Invariant Variational Problems^*

Given a Lie group acting on the space of independent (spacetime) and dependent (field) variables, it is proved that a divergence invariant variational problem is equivalent to a strictly invariant variational problem if and only if a certain associated cohomology class in the invariant variational bicomplex vanishes. This result is illustrated by several examples, starting with the free particle Lagrangian that appeared in Emmy Noether’s original paper, and includes derivations of associated conservation laws through application of her First Theorem. The chapter concludes with some speculations as to the role such cohomology classes might play in fundamental physics, based on the construction of suitable invariant Lagrangians.