Two variational problems in classical differential geometry

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Two variational problems in classical differential geometry

 

Author: Wong, Chong-yung
Title: Two variational problems in classical differential geometry
Degree: M.Sc.
Year: 2000
Subject: Geometry, Differential
Hong Kong Polytechnic University -- Dissertations
Department: Multi-disciplinary Studies
Dept. of Applied Mathematics
Pages: iii, 66 leaves : ill. ; 30 cm
Language: English
InnoPac Record: http://library.polyu.edu.hk/record=b1527694
URI: http://theses.lib.polyu.edu.hk/handle/200/1267
Abstract: This dissertation deals with the following two simple variational problems in classical differential geometry: (a) The Brachistochrone problem on the sphere and the cone (b) Geodesic problems on tubes. Chapters 1 to 3 are devoted to introducing the preliminary results in differential geometry necessary for the discussions of the problems arising in subsequent chapters. In Chapters 4, the classical Brachistochrone problem on some special surfaces is studied. It is shown that the solution of the Brachistochrone problem can be expressed in terms of elliptic integrals. In Chapter 5, geodesics on tubular surfaces such as the torus and the helical tube are studied. Applying the theory of Clairaut relation, we show that the solutions to the geodesic problem over a torus can be fully understood qualitatively. As for the geodesic problem over a helical tube, a careful analysis of the geodesic equations leads to a similar characterization of the solutions. It is hoped that this investigation can serve as simple models to tackle difficult problems in real-life situations, such as determination of yarn shapes and the DNA helix.

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