This article is cited in 4 papers

Volume geodesic distortion and ricci curvature for Hamiltonian dynamics

A. A. Agrachev^ab, D. Barilari^c, E. Paoli<sup>a

a</sup> SISSA, Via Bonomea 265, Trieste (Italy)
^b Steklov Math. Inst., Moscow (Russia)
^c IMJ-PRG, UMR CNRS 7586, Université Paris-Diderot, Batiment Sophie Germain, Case 7012, 75205 Paris Cedex 13 (France)

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, called volume geodesic derivative, 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 asymptotic expansion of the volume. This generalizes the well-known expansion of the Riemannian volume in terms of Ricci curvature to a wide class of Hamiltonian flows, including all sub-Riemannian geodesic flows.

MSC: 53C17, 53B21, 53B15

Received: 18.10.2016
Revised: 15.01.2018
Accepted: 13.03.2018

Language: English

DOI: 10.5802/aif.3268

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ArXiv: 1602.08745