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...Calculus of variations minimal surface revolution siqi clover zheng abstract finding surfaces is a classical problem solved by calcu lus we will rst present catenoid solution using vari ation and then discuss the conditions existence considering maximum separation between two rings finally extend to solving constant mean curvature lagrange multipliers investigating constants involved in dierent delaunay contents introduction euler equation lagrangian continuous discontinuous goldschmidt special case sphere general acknowledgement references eld mathematical analysis that seeks nd path curve etc minimizes or maximizes given function it involves nding u x produces extreme values functional i e denite integral involving its derivative z b f dx brachistochrone often considered birth fastest decent from point lower other problems like hanging cable isoperimetric also employ this paper focus on specically zero extending before diving into some useful tools be introduced let s consider minimi...