Calculus of Variations |
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admissible arc assume Calculus of Variations conjugate point consider CORNELL CORNELL UNIVERSITY CORNELL LIBRARY curve defined dependent variable derivatives DuBois-Reymond dx X1 envelope Euler's equation exist f dx f y'y family of extremals finite function fy'y fy'z fyly fz'z G. A. Bliss geodesic implies integral interval J₁ J₂ least of class lecture Legendre Legendre's condition Legendre's determinant LIBRARY CORNELL UNIVERSITY limit maxima and minima maximum Mayer field minimizing arc E12 necessary condition osculating plane P₂ Proof proper minimum properties quadratic form relations second variation sin²e slope solution of Euler's solved strong relative minimum subsidiary equations sufficient conditions surface Taylor series theorem tremal UNIVE UNIVERSITY LIBRARY CORNELL vanish variational calculus weak relative minimum Weierstrass X₁ Y₁ zero α α ΘΩ ду