Calculus of Variations and Geometric Measure Theory
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A. Agrachev - D. Barilari - E. Paoli

Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics

created by barilari on 28 Feb 2016
modified on 04 Jun 2016

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Submitted Paper

Inserted: 28 feb 2016
Last Updated: 4 jun 2016

Year: 2016

Abstract:

We study the variation of a smooth volume form along extremals of a variational problem with nonholonomic constraints and an action-like Lagrangian. We introduce a new invariant describing the interaction of the volume with the dynamics and we study its basic properties. We then show how this invariant, together with curvature-like invariants of the dynamics, appear in the expansion of the volume at regular points of the exponential map. This generalizes the well-known expansion of the Riemannian volume in terms of Ricci curvature to a wide class of geometric structures, including all sub-Riemannian manifolds.


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