On the analytical solution of the brachistochrone problem in a non-conservative field
type: Item
instance of: scholarly article
title: On the analytical solution of the brachistochrone problem in a non-conservative field
zbMATH Open document ID: 0909.70010
DOI: 10.1016/S0020-7462(97)00026-7
author: B. Vratanar
author: Miran Saje
published in: International Journal of Non-Linear Mechanics
publication date: 1999-04-07T00:00:00Z
review text: We present a new approach to obtain an analytical solution of the brachistochrone problem in a non-conservative velocity-dependent frictional resistance field. Geometrical and energy constraints are incorporated into a time functional through Lagrangian multipliers, and the Euler-Lagrange equations are derived in a natural coordinate system. The novelty of the present approach is a parametrization of the Euler-Lagrange equations by the slope angle of the trajectory. By exploiting a special structure of the governing equations, all function-variables are eliminated and the remaining two unknown parameters are eventually determined from the two nonlinear equations. A numerical example assuming the linear resistance law (Newtonian fluid) is presented, and the influence of the coefficient of viscous friction, \(k\), on the brachistochrone motion is analyzed. The limiting case \(k\to \infty\) is also dealt with.
Mathematics Subject Classification ID: 70F99
Mathematics Subject Classification ID: 49N99
zbMATH DE Number: 1174786
zbMATH Keywords: slope angle of trajectory
zbMATH Keywords: Bernoulli's problem
zbMATH Keywords: Newtonian fluid
zbMATH Keywords: frictional resistance field
zbMATH Keywords: Lagrangian multipliers
zbMATH Keywords: Euler-Lagrange equations
zbMATH Keywords: natural coordinate system
zbMATH Keywords: parametrization
zbMATH Keywords: linear resistance law
zbMATH Keywords: viscous friction
MaRDI profile type: Publication
full work available at URL: s0020-7462(97)00026-7
OpenAlex ID: W2008714703
description: scientific article; zbMATH DE number 1174786