变分法中保持几何意义的一种方法
A Method for Preserving Geometric Meaning in the Calculus of Variations
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中文总结 AI 辅助
该研究提出移动曲面微积分方法，可保留优化问题解的几何意义，避免坐标相关困难，并用于求解最速降线问题，给出曲线的几何表征。
中文摘要 AI 辅助
我们证明，对于源自几何学的优化问题，移动曲面微积分相较于欧拉-拉格朗日方程具有显著优势。作为张量微积分的扩展，它提供了分析相关量的不变表示而非特定坐标表示的工具，这使我们能避免与坐标使用相关的诸多困难，从解析表达式的难以处理的复杂性到恢复最终结果几何解释的几乎不可能。作为示例，我们求解最速降线问题，并根据其曲率给出所求曲线的几何表征。
英文摘要
We discuss the advantages of the Calculus of Moving Surfaces over the Euler-Lagrange equation for optimization problems originating in Geometry. An extension of Tensor Calculus, it provides tools for analyzing geometric quantities directly rather than their coordinate representations. This allows us to avoid the many difficulties associated with the use of coordinates, from the untenable complexity of analytical expressions to the virtual impossibility of recovering the geometric interpretation of the final result. As an illustration, we analyze the brachistochrone and give its geometric characterization in terms of its curvature.
