A unified framework for studying extremal curves on real Stiefel manifolds is presented. We start with a smooth one-parameter family of pseudo-Riemannian metrics on a product of orthogonal groups acting transitively on Stiefel manifolds. In the next step Euler-Langrange equations for a whole class of extremal curves on Stiefel manifolds are derived. This includes not only geodesics with respect to different Riemannian metrics, but so-called quasi-geodesics and smooth curves of constant geodesic curvature, as well. It is shown that they all can be written in closed form. Our results are put into perspective to recent related work where a Hamiltonian rather than a Lagrangian approach was used. For some specific values of the parameter we recover certain well-known results.

中文翻译：

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Stiefel流形上的极值曲线的拉格朗日方法

提出了一个研究Stiefel流形上极值曲线的统一框架。我们从一个光滑的一参数系列伪黎曼度量开始，该系列是在Stiefel流形上传递的正交组的乘积。在下一步中，推导了Stiefel流形上一整类极值曲线的Euler-Langrange方程。这不仅包括关于不同黎曼度量的测地线，而且还包括所谓的准测地线和恒定测地曲率的平滑曲线。结果表明，它们全都可以用封闭形式编写。我们的研究结果被用于最近的相关工作中，其中使用了哈密顿方法而不是拉格朗日方法。对于参数的某些特定值，我们恢复了某些众所周知的结果。