Building on standard Euler-Poincaré 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-Poincaré 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.
A Variational Formulation of Accelerated Optimization on Riemannian Manifolds
最近，Su等人（2016）证明了Nesterov的加速梯度法用于最小化光滑凸函数$f$可以被认为是二阶常微分方程的时间离散化，并且$f(x(t))$沿着该常微分方程的任何轨迹$x(t)$以$\mathcal{O}(1/t^2)$的速率收敛到其最优值。Wibisono等人（2016）引入了一种变分公式，该公式允许在赋范向量空间中以$\mathcal{O}(1/t^p)$的速率加速收敛，对于任意$p>0$。Duruisseaux等人（2021）利用这个框架来设计辛加速优化的有效显式算法。在Alimisis等人（2020）中，提出了一个二阶常微分方程作为黎曼加速算法的连续时间极限，并证明了目标函数$f(x(t))$沿着该常微分方程的解以$\mathcal{O}(1/t^2)$的速率收敛到其最优值。本文通过研究黎曼流形上一类含时Bregman Lagrange系统和哈密顿系统，证明了在黎曼流形上$f(x(t))$到其最优值的收敛速度也可以加速到任意的收敛速度$\mathcal{O}(1/t^p)$.这将Wibisono et al.（2016）的结果推广到了黎曼流形，并为黎曼流形的加速优化提供了一个变分框架。在Duruisseaux et al.（2021）中，基于Bregman Lagrangians和Hamilton族的时间不变性的方法被用于构造非常有效的优化算法，并且我们在黎曼设置中…
A Riemannian Variational and Spectral Framework for High-Dimensional Sphere Packing: Barrier-Dynamics Reconciliation, Periodic Rigidity, and Discrete-Time Guarantees
