jack wrote:

What does it mean to say a field has " an infinite degrees of
freedom"?

it means that the field is an arbitrary element of an infinite-dimensional
manifold, and hence needs infinitely many real numbers for its precise
description (by means of continuous operations).

Now in classical mechanics I understand that the number of
variables in the Lagrangian of the system is not synonomous with the
number of degrees of freedom because one can come up with different
configuation space variables.

Not if you don't allow constraints. If you do allow constraints,
you must subtract the number of degrees of freedoms fixed by the
constraints - then the result is description invariant.

The underlying mathematics is the implicit function theorem,
which gives nondegeneracy conditions under which a n-dimensional manifold
constrained by k conditions results in a (n-k)-dimensional manifold.
(Think of solving systems of linear equations, or of intersecting
3D surfaces. Exclude degenerate situations.)


Arnold Neumaier