The variational structure of the space of holonomic measures

Roughly speaking, holonomic measures are parametric varifolds without boundary. They provide a setting appropriate for the analysis of many variational problems. In this paper, we characterize the space of variations for these objects, and we use the characterization to formulate stability conditions that are strictly more general than the Euler-Lagrange equations. We also use this characterization to deduce higher-dimensional analogues of energy conservation and weak KAM. Along the way, we characterize the distributions that arise as derivatives of families of Borel probability measures on smooth manifolds.