Optimal Paths for Symmetric Actions in the Unitary Group

Abstract

Given a positive and unitarily invariant Lagrangian \({\mathcal{L}}\) defined in the algebra of matrices, and a fixed time interval \({[0,t_0]\subset\mathbb R}\), we study the action defined in the Lie group of \({n\times n}\) unitary matrices \({\mathcal{U}(n)}\) by $$\mathcal{S}(\alpha)=\int_0^{t_0} \mathcal{L}(\dot\alpha(t))\,dt, $$where \({\alpha:[0,t_0]\to\mathcal{U}(n)}\) 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 π. Moreover, if \({\mathcal{L}}\) 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.