Integrable sub-Riemannian geodesic flows on the special orthogonal group

TL;DR

This paper demonstrates that most normal geodesics in a specific sub-Riemannian structure on the special orthogonal group are integrable, using a method adapted from rigid body dynamics involving Lax pairs and bi-Hamiltonian structures.

Contribution

It establishes the complete integrability of almost all geodesics in a sub-Riemannian setting on SO(n), extending techniques from rigid body mechanics.

Findings

Most geodesics are completely integrable.

Lax pair and bi-Hamiltonian structures are constructed.

Method adapts rigid body integrability techniques.

Abstract

We analyse the geometry of the rubber-rolling distribution on the special orthogonal group and show that almost all the normal geodesics of any right-invariant sub-Riemannian metric defined on this distribution are completely integrable. Our argument is an adaptation of the method used to establish integrability of the Riemannian metric arising from the $n$-dimensional rigid body: namely, by exhibiting a Lax pair and bi-Hamiltonian structure for the reduced equations of motion.