Lie Algebras and Integrable Systems: Elastic Curves and Rolling Geodesics

This paper follows a long-standing fascination in the relevance of Lie algebras and Lie groups for problems of applied mathematics. It originates with the discovery that the mathematical formalism initiated by G. Kirchhoff to model the equilibrium configurations of an elastic rod can be extended to the isometry groups of certain Riemannian manifolds through control theoretic insights and the Maximum Principle, giving rise to a large class of Hamiltonian systems that link geometry with physics in novel ways. This paper focuses on the relations between the Kirchhoff-like affine–quadratic problem and the rolling geodesic problem associated with the rollings of homogeneous manifolds $G/K$, equipped with a $G$-invariant metric, on their tangent spaces. We will show that there is a remarkable connection between these two problems manifested through a common isospectral curve in the Lie algebra $\mathfrak g$ of $G$. In the process we will reveal the significance of curvature for the theory of elastic curves.