Field theory is one of the cornerstones of classical physics. The most notable
examples of classical fields are the force fields that one encounters in the description
of gravitational and electromagnetic phenomena. These fields are
caused by the presence of masses and electric charges, respectively. In this
chapter we present a framework for classical field theory, which is known as
the Lagrangian formulation. In this formulation the dynamics of a system is
described by a single function, the Lagrangian. Via a variational principle the
Lagrangian yields the equations of motion which govern the time evolution of
that system, so it is a useful mnemonic for summarizing a theory in a concise
form. The use of a variational principle to express the equations of classical
physics is very old. Fermat’s principle in optics (1657) and Maupertuis’
principle in mechanics (1744) are famous examples. 

Apart from its conciseness and its mathematical elegance we mention two
important reasons why the Lagrangian formulation is so convenient in field
theory. The first one is that the Lagrangian, or rather the integral of the
Lagrangian density over space and time, should be invariant under all symmetries
of the theory in question. This aspect of the Lagrangian formulation
makes it rather attractive for relativistic theories, because it allows one to
treat space and time on an equal footing from the very beginning, in contrast
to other approaches where one aims directly for a description of the time evolution.
The second advantage of the Lagrangian formulation, emphasized by
Dirac and elaborated on by Feynman, is that it plays a natural role in the
path integral formulation of quantum mechanics. It turns out that the evolution
operator for a quantum-mechanical wave function can be expressed as
a sum over all paths with fixed endpoints in space-time weighted by a phase