A New Theory of Parametric Problems in the Calculus of Variations ... |
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2n-2)-parameter family Amer arc E/2 arguments of f Calculus of Variations canonical differential equations conjugate points constant ct+d defining the transversal derivatives different from zero DOCTOR OF PHILOSOPHY Dubois-Reymond equations 2-8 equations 6-13 equations y)=0 Euler differential equations existence theorems extremal arc f₁ family of extremals Fixx follows FRANK LYNWOOD WREN function W(x fundamental system Furthermore G.A.Bliss Hamilton-Jacobi partial differential Hamilton-Jacobi theory Hamiltonian Hence identically zero initial values Jacobi condition Jacobi equations Jacobi's necessary condition Legendre's quadratic form Lemma linearly independent Math matrix McFerrin School minimizing arc obtain original problem orthogonal variation parameter PARAMETRIC PROBLEMS partial differential equation points 3 conjugate Professor Bliss region F satisfies the Hamilton-Jacobi satisfy the relation second variation slope-functions p(x t₁ t₂ Tennessee Theorem XII THEORY OF PARAMETRIC throughout F tions University of Chicago variations in parametric Weierstrass E-function x²(t бра