Every physics student knows that the energy of an
electromagnetic field is equal to $1/2(E^2+B^2)$. However, there are a lot
of ways to obtain this formula. Some lecturers use heuristic arguments
about transferring charged particles from infinity. Another idea is to
derive it from the energy-momentum tensor - then the canonical
energy-momentum tensor must be artificially improved by
Belinfante-Rosenfeld procedure to obtain the desired formula. However,
these derivations are not so satisfying from the formal and physical
point of view. I will show how to obtain this formula as a generating
function of dynamics using symplectic formalism proposed by W. M.
Tulczyjew in a paper A Symplectic Framework of Linear Field Theories.
First, I will define the energy of a field in every linear field theory
using symplectic space associated with current given by the
three-dimensional volume $V$ and transversal time vector field on it.
Next, I will show how to apply this construction to Maxwell
electrodynamics, and how it can be useful in proving the existence of a
solution of the initial-boundary problem by evolutionary methods.