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Field theory

Let us continue our visit to classical mechanics. We have discussed the action principle as it applies to a single particle moving in one and three dimensions. Now consider the mechanics of a violin string, which is allowed to vibrate up and down (but not sideways) while tied at its ends. An element of the string is denoted by its coordinate x along the string, and in its motion it is displaced in a direction transverse to the string by a distance tex2html_wrap_inline1062 . Now the function tex2html_wrap_inline1064 constitutes an infinite set of coordinates for which x is the index. (Recall the correspondence tex2html_wrap_inline1068 in Section 1.) A function tex2html_wrap_inline1062 of space and time is a trajectory of the entire string. It is not hard to show that the kinetic and potential energies of the string are given by

eqnarray340

where tex2html_wrap_inline1072 is the length of the string; tex2html_wrap_inline1074 is its density (mass per unit length); and tex2html_wrap_inline1076 is the tension of the string. As before L=T-V, and thus the action is

eqnarray348

This is of the form

equation356

and so the Euler-Lagrange equation is [cf. (28)]

eqnarray360

This partial differential equation is the wave equation,

equation378

where we identify the wave velocity as tex2html_wrap_inline1080 .

The string is the simplest example of a problem in continuum dynamics, also called classical field theory. tex2html_wrap_inline1062 is the field, a variable associated with each point in space that evolves dynamically in time. (Think of the electric field, governed by Maxwell's Equations.) S is the action for the field theory, and the wave equation is the equation of motion, which is a partial differential equation because of the added element of x-dependence.

For any trajectory tex2html_wrap_inline1062 , the energy is the sum of the kinetic and potential energies,

eqnarray390

Note that there is no integral over time in defining the energy. One might be interested in finding the time-independent shape of the string that minimizes the energy. Here the kinetic energy is zero, and one minimizes the functional

  equation398

The Euler equation here is

equation404

which gives the trivial solution (satisfying the boundary conditions) tex2html_wrap_inline1090 . This is also obvious from the form of (50). It is of course easy to invent an energy-minimization problem with a non-trivial solution.gif The point of this exercise is to show that the same problem might invite both time-dependent and time-independent variational calculations.

Of course, not all field theory problems are one-dimensional like the vibrating string. Examples of two-dimensional field theory are the vibrations of a drumhead or surface waves in water, where the physics is described by a function tex2html_wrap_inline1096 and an action functional tex2html_wrap_inline1098 . Electromagnetism is a three-dimensional field theory, with six separate fields that are the components of tex2html_wrap_inline1100 and tex2html_wrap_inline1102 .


next up previous
Next: Constraints Up: Functionals Previous: Minimization and classical mechanics