Calculus of Variations |
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a₁ arbitrary arc length brachistochrone problem C₁ Calculus of Variations Chapter circle class C₂ class D₁ consider constant corner conditions cosh curve joining curve of class cycloids defined differential dx dy dx₁ dy₁ end-point conditions Euler's equation Euler's rule example extremal arc extremal joining F₁ field of extremals geodesic gives Hence integral of Euler's integrand isoperimetrical problem joining the end-points line joining minimising curve one-parameter family ordinary problem P₁ and P₂ parabola parameter parametric representation positive-homogeneous of degree satisfied semi-field sinh slope function solid of revolution solution stationary value straight line strong relative minimum sufficiently small surface of revolution t₁ t₂ unique extremal variable end-point theorem varied curve weak relative minimum whence write x-axis x₁ y₁ zero απ θα г₁ г₂ ди дн дх дх