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 defining the transversal derivatives different from zero Dubois-Reymond E₁₂ equations 2-8 equations 6-13 equations K()=0 Euler differential equations EULER EQUATIONS existence theorems extremal arc f₁ family of extremals faxi Fixx follows 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 k₁ Legendre's quadratic form Lemma linearly independent Math Mathematics matrix McFerrin School minimizing arc obtain original problem orthogonal solution ult orthogonal variation parameter partial differential equation points 3 conjugate Professor Bliss region F satisfies the Hamilton-Jacobi satisfy the relation second variation slope-functions p(x SOLUTIONS OF JACOBI solutions uk t₁ t₂ Theorem XII throughout F tions U₁ University of Chicago variations in parametric Weierstrass E-function