  Starting with a functional difficulty appeared in the paper \cite{vz00} by
V\'azquez and Zuazua, we obtain new insights into the Hardy Inequality and the
evolution problem associated to it by means of a reformulation of the problem.
Surprisingly, the connection of the energy of the new formulation with the
standard Hardy functional is nontrivial, due to the presence of a Hardy
singularity energy. This corresponds to a loss for the total energy. The
problem arises when the equation is posed in a bounded domain, and also when
posed in the whole space.
  We also consider an equivalent problem with inverse square potential on an
exterior domain. The extra energy term is then present as an effect that comes
from infinity, a kind of hidden energy. In this case, in an unexpected way,
this term is additive to the total energy, and it may even constitute the main
part of it.
