Lecture 10

Handouts:
7. Homework 2 Solutions
8. Writing Assignment 1

Today we tried to relate classical field equations (such as Maxwell's
equations) to an action principle, hoping to reproduce the
correspondence that occurs in classical particle mechanics between
Newton's second law and Hamilton's principle.

For simplicity, we considered a scalar field \phi (scalar under
Lorentz transformations) that obeys a wave equation. We showed how
such a field could arise from Newtonian mechanics, applied to a
continuous medium.

We considered the motion of a string. Such a string has constant
tension (or energy per unit length), so that its equilibrium position
in space is a straight line. Deviations from equilibrium experience a
spring-like restoring force. We considered deviations from
equilibrium, along the length of the string. We approximated the
equilibrium string as a discrete series of massive particles, each
separated by a grid step d. We let each particle (indexed by j) be
displaced from its equilibrium position by q_j, then calculated the
restoring force on particle j due to the springlike interactions with
its nearest neighbors j-1 and j+1. We found the equation of motion for
q_j (given by Newton II), as well as the kinetic and potential energy
of the string.

We then took the continuum limit of our discrete approximation: we let
the grid spacing d go to zero, and the number of point masses n in the
grid go to infinity, while keeping the following *physical*
characteristics of the string constant: the length nd, the tension
m/d, and the local spring force, proportional to kd. The idea here is
that our limit doesn't vary the physical string we are considering,
just the fineness of our approximation to it. This limit gave us: 1) a
one-dimensional wave equation for the displacement q(x,t); 2) a
kinetic energy density proportional to \dot{q}^2; and 3) a potential
energy density proportional to (\partial_x q )^2. Note that springlike
interactions produce gradient^2 terms in the potential energy density,
in the continuum limit. We obtained a Lagrangian density proportional
to 1/c^2 \dot{q}^2 - (\partial_x q)^2.

Had we considered a 3-dimensional continuum of particles, with
displacement \phi in any uniform fixed direction, we would have
obtained a full-fledged 3-dimensional wave equation for \phi, with the
Lagrangian being a space-volume integral of the Lorentz-invariant form
\partial_\mu \phi \partial^\mu \phi. 

Note that, since we have derived the field equation for \phi from
Newton's second law, it must be equivalent to the Euler-Lagrange
equations for the calculated Lagrangian. Thus it must be equivalent to
extremizing the action, or time-integral of the Lagrangian. Since the
Lagrangian density is Lorentz invariant, and the spacetime volume
element dt d^3 x is Lorentz invariant, the action itself is Lorentz
invariant. Thus the field equations are equivalent to the following
Lorentz invariant statement: the field evolves so as to extremize the
action, which is Lorentz-invariant.

Not all fields have pre-existing Newtonian analogues, as they describe
interactions more novel than the displacement-related ones above. For
such fields, and for brevity, we would like to know how to find the
Euler-Lagrange field equations directly, without discretizing the
system. We use the simple example above as a model for building the
Lagrangians and Euler-Lagrange equations for more general scalar field
theories, next time.

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KB