Sub-Riemannian cubics in SU(2)

TL;DR

This paper explores sub-Riemannian cubics within the Lie group SU(2), focusing on their properties and long-term behavior, extending the concept of Riemannian cubics to a constrained sub-Riemannian setting.

Contribution

It introduces and analyzes sub-Riemannian Lie quadratics in SU(2), highlighting their dynamics and extending the theory of cubics to sub-Riemannian Lie groups.

Findings

Characterization of sub-Riemannian Lie quadratics in SU(2)

Analysis of long-term dynamics of these curves

Extension of cubic curve theory to sub-Riemannian Lie groups

Abstract

Sub-Riemannian cubics are a generalisation of Riemannian cubics to a sub-Riemannian manifold. Cubics are curves which minimise the integral of the norm squared of the covariant acceleration. Sub-Riemannian cubics are cubics which are restricted to move in a horizontal subspace of the tangent space. When the sub-Riemannian manifold is also a Lie group, sub-Riemannian cubics correspond to what we call a sub-Riemannian Lie quadratic in the Lie algebra. The present article studies sub-Riemannian Lie quadratics in the case of $\mathfrak{su}(2)$, focusing on the long term dynamics.