Optimal paths for symmetric actions in the unitary group

TL;DR

This paper characterizes optimal paths in the unitary group under a unitarily invariant Lagrangian, showing that one-parameter subgroups are optimal and unique under certain conditions, with implications for metrics on unitary and Grassmann manifolds.

Contribution

It proves that one-parameter subgroups are the optimal paths for a class of unitarily invariant Lagrangians, establishing their uniqueness when the Lagrangian is strictly convex.

Findings

One-parameter subgroups are optimal paths under the given Lagrangian.

Optimality is guaranteed when the spectrum of the exponent is bounded by π.

Uniqueness of optimal paths is established for strictly convex Lagrangians.

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