  This article aims to study the Lorentz/U(1) gauge symmetry-breaking
electrodynamics in the framework of the Standard-Model Extension and analyze
the Hamiltonian structure for the theory with a specific dimension $d\leq 4$ of
Lorentz breaking operators. For this purpose, we consider a general quadratic
action of the modified electrodynamics with Lorentz/gauge-breaking operators
and calculate the number of independent components of the operators at
different dimensions in gauge invariance and breaking. With this general
action, we then analyze how the Lorentz/gauge symmetry-breaking can change the
Hamiltonian structure of the theories by considering Lorentz/gauge-breaking
operators with dimension $d\leq 4$ as examples. We show that the
Lorentz-breaking operators with gauge invariance do not change the classes of
the constraints of the theory and the number of the physical degrees of freedom
of the standard Maxwell's electrodynamics. When the U(1) gauge
symmetry-breaking operators are present, the theories in general lack
first-class constraint and have one additional physical degree of freedom,
compared to the standard Maxwell's electrodynamics.
