ABSTRACT
Classical field theory is a generalization of classical mechanics, which deals with discrete particles and a finite number of variables, to situations where one has to consider continuous matter and correspondingly an infinite number of variables. This is actually quite familiar to all who have studied electromagnetism. Although one starts with charged particles, it soon becomes necessary to consider electric fields and magnetic fields which are extended over space. There are equations governing the evolution with time of such fields. The variables of this subject then are the electric and magnetic fields, holding an infinite number of degrees of freedom. Instead of having Lagrangians and Hamiltonians depending on a finite number of generalized coordinates and momenta, one clearly needs a continuum of coordinates and likewise a continuum of momenta. Thus, instead of variables like q(t), one needs variables
like ~E(~x, t). Lagrangians too have to depend on such fields. One thus comes to quantities which depend on functions of space coordinates, in addition to the more common time coordinate.
