Deformations of closed measures and variational characterization of measures invariant under the Euler-Lagrange flow

TL;DR

This paper develops a variational calculus framework for closed measures, characterizes their derivatives, and links critical measures to Euler-Lagrange invariance, providing insights into minimizers in calculus of variations.

Contribution

It offers a full description of derivatives of closed measures and characterizes those invariant under the Euler-Lagrange flow, advancing the understanding of variational problems.

Findings

Characterization of derivatives of closed measures.

Conditions under which critical measures satisfy Euler-Lagrange equations.

Proof that all minimizers are invariant under the Euler-Lagrange flow.

Abstract

The set of closed (or holonomic) measures provides a useful setting for studying optimization problems because it contains all curves, while also enjoying good compactness and convexity properties. We study the way to do variational calculus on the set of closed measures. Our main result is a full description of the distributions that arise as the derivatives of variations of such closed measures. We give examples of how this can be used to extract information about the critical closed measures. The condition of criticality with respect to variations leads, in certain circumstances, to the Euler-Lagrange equations. To understand when this happens, we characterize the closed measures that are invariant under the Euler-Lagrange flow. Our result implies Ricardo Ma\~n\'e's statement that all minimizers are invariant.