A new approach to brachistochrone problem with an account dry friction

Abstract

Background. The study proposes a new variational approach to solving the problem of the brachistochrone the purpose of which is to mathematically rigorously substantiate all the basic equations of the dynamic motion of a body in a mobile basis. The relevance of the research topic is dictated primarily by the novelty of the task and the methodology of its solution. Materials and methods. The solution method is based on the use of a moving basis and a variational approach. Results. A general fundamental solution of the twodimensional Laplace equation for a function dependent on three independent coordinates is obtained. Conclusions. A system of differential equations strictly substantiated with the help of the variational approach describing the optimal trajectory of the body's motion in a mobile basis has been obtained.

Keywords

• brachistochrone
• dry friction
• equation of motion
• variational principle