Variational Lifting and Optimal Gauges on Riemannian Homogeneous Spaces

Abstract

Building on standard Euler-Poincar\'e reduction on Lie groups, we develop a variational lifting framework for mechanical systems on Riemannian homogeneous spaces $H=G/K$. A mechanical action on $H$ is lifted to $G$ through a functional whose kinetic energy depends only on the horizontal component of the velocity, and we prove that its critical points project precisely onto those of the original action. The lifted functional possesses a natural gauge invariance under $H^1$ curves with values in the isotropy subgroup $K$. Consequently, every lifted critical curve decomposes into a smooth horizontal representative and an arbitrary vertical gauge, and the associated Euler-Poincar\'e equations with symmetry breaking are obtained in reduced form. We then introduce a second variational problem that selects a distinguished lift by minimizing the vertical kinetic energy. The optimal gauge is a length-minimizing geodesic on $K$, yielding an explicit decomposition of the total energy into projected and vertical contributions and a geometric interpretation in terms of holonomy for closed projected curves. The framework is illustrated first for pure quantum states on $\mathbb{CP}^{n}\cong SU(n+1)/S(U(1)\times U(n))$, where the gauge freedom corresponds to the isotropy of unitary lifts, and then for $S^2\cong SO(3)/SO(2)$ through an optimal camera-orientation problem for an axisymmetric satellite subject to an undesirable pointing region.