Introductory note to s1899b

pp. 280-285

in: Ernst Zermelo, Calculus of variations, applied mathematics, and physics/Variationsrechnung, angewandte mathematik und physik, Berlin, Springer, 2013

Abstract

Zermelo treats the problem of motion of a string in a potential field W with the help of Hamilton's principle. A string is an elastic physical body with small cross section. It can be represented by a continuous curve r = r(t) = (x(t), y(t), z(t)) that can assume every possible position. A string is therefore completely flexible, but by assumption inextensible. For inextensible strings that are fixed at both ends, the equations of motion (the Euler equations) become L = (T − U) + λ(S − 1). The Lagrange multiplier λ can be physically interpreted as the tension of the string.