On curvature-type invariants for natural mechanical systems on sub-Riemannian structures associated with a principle G-bundle

Abstract

The Jacobi curve of an extremal of an optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. For parametrized curves in Lagrange Grassmannians satisfying very general assumptions, the canonical bundle of moving frames and the complete system of symplectic invariants, called curvature maps, were constructed. The structural equation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the normal form for the Jacobi equation along this extremal and the curvature maps can be seen as the “coefficients” of this normal form. In the present paper, we focus on the curvature maps for an optimal control problem of a natural mechanical system on a sub-Riemannian structure on a principle connection of a principle G-bundles with one dimensional fibers over a Riemannian manifold. We express the curvature maps in terms of the curvature tensor of the base Riemannian manifold and the curvature form and the potential.