Abstract
The most famous classical variational principle is the so-called Brachistochrone problem. In this work, Homotopy perturbation method (HPM) is applied to the Brachistochrone problem that arises invariational problems. The results reveal the efficiency and the accuracy of the proposed method. Homotopy perturbation method yields solutions in convergent series forms with easy computation.
Highlights
Minimization problems that can be analyzed by the calculus of variations serve to characterize the equilibrium configurations of almost all continuous physical systems, ranging through elasticity, solid and fluid mechanics, electro-magnetism, gravitation, quantum mechanics, string theory, and many other topics
The goal is to shape the wire in such a way that, starting from rest, the bead slides from one end to the other in minimal time
The Brachistochrone problem was originally posed by the Swiss mathematician Johann Bernoulli in 1696, and served as an inspiration for much subsequent development of this subject
Summary
Minimization problems that can be analyzed by the calculus of variations serve to characterize the equilibrium configurations of almost all continuous physical systems, ranging through elasticity, solid and fluid mechanics, electro-magnetism, gravitation, quantum mechanics, string theory, and many other topics. Some of geometrical configurations such as minimal surfaces, can be conveniently formulated as optimization problems. Numerical approximations to the equilibrium solutions of such boundary value problems are based on a nonlinear finite element approach that reduced the infinite dimensional minimization problem to a finite-dimensional one, to which we can apply the optimization techniques. The goal is to shape the wire in such a way that, starting from rest, the bead slides from one end to the other in minimal time. Let us consider the following non-linear functional equation:. Let’s consider the solution of Eq (5) as a power series over p, as follows: The approximate solutions of Eq(1) can be obtained by letting p→1 u = limpp→1 vv = vv0 + vv1 + vv2 + ⋯.
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