Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics

TL;DR

This paper introduces a new invariant related to volume variation in Hamiltonian dynamics with constraints, generalizing Riemannian volume expansion to sub-Riemannian and other geometric structures.

Contribution

It presents a novel invariant that captures volume and curvature interactions in constrained Hamiltonian systems, extending classical Riemannian results.

Findings

The invariant describes volume interactions with dynamics.

Curvature-like invariants appear in volume expansion.

Generalization to sub-Riemannian manifolds.

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.