Belov, V. M.; Rybal’chenko, T. A.; Sukhanov, V. A. Gradient methods for the numerical solution of the problem of the variational calculus in the classical formulation. (English. Russian original) Zbl 0832.65060 Comput. Math. Math. Phys. 34, No. 5, 679-686 (1994); translation from Zh. Vychisl. Mat. Mat. Fiz. 34, No. 5, 784-792 (1994). Summary: Using one of two auxiliary quadratic variational problems with homogeneous boundary conditions, a search for the direction of descent is made and a computational scheme of Newton’s method in the classical formulation is given. It is shown that the transformation of Euler’s gradient by the inverse of the second derivative is equivalent to the method of gradient descent in the modern formulation. MSC: 65K10 Numerical optimization and variational techniques 90C99 Mathematical programming 49J15 Existence theories for optimal control problems involving ordinary differential equations Keywords:brachistochrone problem; quadratic variational problems; Newton’s method; method of gradient descent PDFBibTeX XMLCite \textit{V. M. Belov} et al., Comput. Math. Math. Phys. 34, No. 5, 679--686 (1994; Zbl 0832.65060); translation from Zh. Vychisl. Mat. Mat. Fiz. 34, No. 5, 784--792 (1994)