Брахистохрона и верижна линия

This paper is didactic in character, with an emphasis on the history of the problems discussed. The classical brachistochrone problem is considered: finding the curve along which a particle, moving without friction under gravity, travels from one given point to a lower one in the shortest possible time. The problem is reduced to the minimization of a functional and solved by means of the Euler–Lagrange equation in the form of the Beltrami identity. The required curve turns out to be a cycloid. The connection with Fermat’s principle and the tautochrone property of the cycloid are discussed. The same method is then applied to the isoperimetric problem of the hanging chain: minimizing the potential energy at a prescribed length leads to the catenary.Keywords: calculus of variations; Euler–Lagrange equation; brachistochrone; isoperimetric problem; catenary.