  We consider a non-negative and one-homogeneous energy functional $\mathcal J$
on a Hilbert space. The paper provides an exact relation between the solutions
of the associated gradient-flow equations and the energetic solutions generated
via the rate-inpendent system given in terms of the time-dependent functional
$\mathcal E(t,u)=t \mathcal J(u)$ and the norm as a dissipation distance. The
relation between the two flows is given via a solution-dependent
reparametrization of time that can be guessed from the homogeneities of energy
and dissipations in the two equations. We provide several examples including
the total-variation flow and show that equivalence of the two systems through a
solution dependent reparametrization of the time. Making the relation
mathematically rigorous includes a careful analysis of the jumps in energetic
solutions which correspond to constant-speed intervals for the solutins of the
gradient-flow equation. As a major result we obtain a non-trivial existence and
uniqueness result for the energetic rate-independent system.
