Rolling on Affine Tangent Planes: Parallel Transport and the Associated Sub-Riemannian Problems

Abstract

This paper addresses the natural optimal control problem associated with the kinematic equations generated by the rollings of symmetric Riemannian spaces on their affine tangent spaces at a fixed point. This optimal problem can be viewed as a certain left-invariant sub-Riemannian problem on a Lie group G associated with a three-step distribution $\mathcal {F}$ on the Lie algebra $\mathfrak {g}$ of G. We will use the Maximum Principle of optimality to obtain the appropriate Hamiltonian, and then we will show that the corresponding Hamiltonian system of equations admits an isospectral representation, and hence, is completely integrable. As a byproduct, this representation reveals intriguing connections with mechanical tops, while at the same time, it sheds additional light on the discovery in [7] that the elastic curves, elasticate in Euler’s terminology, can be obtained entirely by the rolling sphere problems on spaces of constant curvature.