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We consider a non-negative and one-homogeneous energy functional $\mathcal J$on a Hilbert space. The paper provides an exact relation between the solutionsof the associated gradient-flow equations and the energetic solutions generatedvia 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. Therelation between the two flows is given via a solution-dependentreparametrization of time that can be guessed from the homogeneities of energyand dissipations in the two equations. We provide several examples includingthe total-variation flow and show that equivalence of the two systems through asolution dependent reparametrization of the time. Making the relationmathematically rigorous includes a careful analysis of the jumps in energeticsolutions which correspond to constant-speed intervals for the solutins of thegradient-flow equation. As a major result we obtain a non-trivial existence anduniqueness result for the energetic rate-independent system.

Relating a rate-independent system and a gradient system for the case of one-homogeneous potentials