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We study the variation of a smooth volume form along extremals of avariational problem with nonholonomic constraints and an action-likeLagrangian. We introduce a new invariant describing the interaction of thevolume with the dynamics and we study its basic properties. We then show howthis invariant, together with curvature-like invariants of the dynamics, appearin the expansion of the volume at regular points of the exponential map. Thisgeneralizes the well-known expansion of the Riemannian volume in terms of Riccicurvature to a wide class of geometric structures, including all sub-Riemannianmanifolds.

Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics