Elementary Problems for Elastic Strings

Abstract

The development of both continuum mechanics and mathematics during the eighteenth century was profoundly influenced by the successful treatment of conceptually simple, but technically difficult problems. (Indeed, one can argue that the dominant philosophical attitudes of the Age of Reason were founded on an awareness of these scientific triumphs, if not on their understanding.) Among the most notable of these classical problems are those of determining the equilibrium states of inextensible strings hung between two points and subjected to various systems of loads. (An inextensible string is one for which the stretch v is constrained to equal 1, no matter what force system is applied to the string.) The problem of the catenary is to determine the equilibrium states of such a string when the applied force is the weight of the string. The problem of the suspension bridge is to determine these states when the applied force is a vertical load of constant intensity per horizontal distance. (The string does not correspond to the bridge, but to the wires from which it is suspended.) The problem of the velaria is to determine these states when the applied force is a normal pressure of constant intensity. In the related problem of the lintearia, the applied force is a normal pressure varying linearly with depth. (This problem describes the deformation of a cylindrical membrane holding a liquid, the string representing a typical section of the membrane.) In a fifth problem, the applied force is the attraction to a fixed point. In Sec. 8 we outline the progression from haphazard conjecture to elegant solution of these problems in the seventeenth and eighteenth centuries.