This paper starts with the derivation of the most general equations of motion for the infinitesimal rotators moving on arbitrary two‐dimensional surfaces of revolution. Both geodesic and geodetic (i.e., without any external potential) equations of motion on surfaces with nontrivial curvatures that are embedded into the three‐dimensional Euclidean space are discussed. The Mylar balloon as a concrete example for the application of the scheme was chosen. A new parameterization of this surface is presented, and the corresponding equations of motion for geodesics and geodetics are expressed in an analytical form through the elliptic functions and elliptic integrals. The so‐obtained results are also compared with those for the two‐dimensional sphere embedded into the three‐dimensional Euclidean space for which it can be shown that the geodesics and geodetics are plane curves realized as the great and small circles on the sphere, respectively.

中文翻译：

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聚酯薄膜气球上无穷小旋转子的经典运动

本文首先推导了在任意二维旋转面上运动的无穷小转子的最通用运动方程。讨论了嵌入到三维欧几里得空间中的具有非平凡曲率的表面上的测地线和大地测量（即，没有任何外部电势）运动方程。选择了Mylar气球作为该方案应用的具体示例。给出了该表面的新参数化，并通过椭圆函数和椭圆积分以解析形式表示了大地测量学和大地测量学的相应运动方程。