Departmental Bulletin Paper 曲面上の測地線としての最速降下曲線

森田, 正亮  ,  もりた, まさあき  ,  Morita, Masaaki  ,  沖縄工業高等専門学校 総合科学科

We consider whether a brachistochrone curve, which is known as a trajectory on which a particle in a potential travels in a shortest time between given two points, can be regarded as a geodesic in a surface. We immediately find that a brachistochrone curve is equivalent to a geodesic in a two-dimensional Riemannian manifold with a conformally flat metric, but it is non-trivial whether the manifold can be embedded in a three-dimensional Euclidean space. We attempt to constitute a surface of revolution that realizes the Riemannian manifold, and clarify the relation between the mechanical potential and the surface constituted. We also discuss the possibility that the Riemannian manifold is embedded as a minimal surface.

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