Classical Mechanics (Goldstein) · Problem 1

Introduction to Lagrangian and Hamiltonian Formulations for Continuous Systems — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · lagrangian-continuous-systems

Problem

(a) The transverse vibrations of a stretched string can be approximated by a discrete system consisting of equally spaced mass points located on a weightless string. Show that if the spacing is allowed to go to zero, the Lagrangian approaches the limit L=12∫[μη˙2−T(∂η∂x)2]dxL = \frac{1}{2}\int\left[\mu\dot{\eta}^2 - T\left(\frac{\partial\eta}{\partial x}\right)^2\right]dx for the continuous string, where TT is the fixed tension. What is the equation of motion if the density μ\mu is a function of position? (b) Obtain the Lagrangian for the continuous string by finding the kinetic and potential energies corresponding to transverse motion. The potential energy can be obtained from the work done by the tension force in stretching the string in the course of the transverse vibration.

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