Calculus of Variations by Parallel Displacement

angle axis c₁ C₂ calculus of variations catenary Chapter components conjugate points constant coordinates Coriolis corner condition curve cycloid derived Hilbert integrals differential equations directional derivative dx dx E-function envelope envelope theorem Euler's condition extremal paths extremum family of extremals family of Hilbert family parameter field of extremals field parameter fixed point fundamental identities geodesic given Hamilton's Hamilton's Principle hence integral sign integration path J₁ J₂ Jacobi's Theorem line integral minimizing minimum negative Newton's parallel displacement partials with respect path of integration plane point function positive problem quadratic rates of change shown in Figure single-parameter family slope solution surface surface of revolution taken with respect tangent theorem 12 total derivative transversality condition upper limit vanishing area derivatives variable of integration variations integral vector integrand velocity Weierstrass whereby y²² ӘЈ ән ду дх