Optimal paths for symmetric actions in the unitary group

Abstract

Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval $[a,b]\subset\mathbb R$, we study the action defined in the Lie group of $n\times n$ unitary matrices $\mathcal{U}(n)$ by $$ S(\alpha)=\int_a^b L(\dot\alpha(t))\,dt\,, $$ where $\alpha:[a,b]\to\mathcal{U}(n)$ is a rectifiable curve. We prove that the one-parameter subgroups of $\mathcal{U}(n)$ are the optimal paths, provided the spectrum of the exponent is bounded by $\pi$. Moreover, if L is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in $\mathcal{U}(n)$ as well as angular metrics in the Grassmann manifold