9 - CONTINUUM DYNAMICS

Summary

CHAPTER OVERVIEW

The Lagrangian and Hamiltonian formalisms of particle dynamics can be generalized and extended to describe continuous systems such as a vibrating rod or a fluid. In such systems each point x, influenced by both external and internal forces, moves independently. In the rod, points that start out very close together always remain close together, whereas in the fluid they may end up far apart. The displacement and velocity of the points of the system are described by functions ψ(x, t) called fields.

In this chapter we describe the classical (nonquantum) theory of fields. Particle dynamics will be transformed to field theory by allowing the number of particles to increase without bound while their masses and the distance between them go to zero in such a way that a meaningful limit exists. This is called passing to the continuum limit.

LAGRANGIAN FORMULATION OF CONTINUUM DYNAMICS

PASSING TO THE CONTINUUM LIMIT

THE SINE–GORDON EQUATION

In this section we present an example to show how to pass to the continuum limit. Consider the system illustrated in Fig. 9.1, consisting of many simple plane pendula of length l and mass m, all suspended from a horizontal rod with a screw thread cut into it. The planes of the pendula are perpendicular to the rod (only two of the planes are drawn in the figure), and the pendula are attached to (massless) nuts that move along the rod as the pendula swing in their planes [for more details, see Scott (1969)].