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General Formulation of the Problem
The Fundamental Lemma and Eulers Differential
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abscissa according admissible curve American Mathematical Society assumptions Calculus of Variations co-ordinates coincide Compare conjugate point consider constant continuous corresponding curve of class defined definite integral denote determine different from zero domain E-function Euler's differential equation Euler's equation Example exists extremum finite number footnote function F furnishes geodesic geometrical given points H. A. Schwarz Hence it follows Hence we obtain Hilbert's inequality integral J taken interior interval isoperimetric isoperimetric problem Kneser Lectures Lehrbuch lemma lies entirely limit Mathematische Annalen maxima and minima method minimize the integral minimizing curve minimum Moreover necessary condition neighborhood ordinary curve Osgood parameter parameter-representation partial derivatives positive tangent proof proved rectifiable curve region result satisfied second variation set of extremals sinh straight line Sturm's theorem sufficient conditions sufficiently small suppose theory tion transversal vanish at x0 vicinity Weierstrass's theorem