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We provide several examples of dissipative systems that can be obtained fromconservative ones through a simple, quadratic,change of time. A typical exampleis the curve-shortening flow in R^d, which is a particular caseofmean-curvature flow with co-dimension higher than one (except in the cased=2).Through such a change of time, this flow can be formally derived from theconservative model of vibrating strings obtainedfrom the Nambu-Goto action.Using the concept of "relative entropy" (or "modulated energy"), borrowed fromthe theoryof hyperbolic systems of conservation laws, we introduce a notion ofgeneralized solutions,that we call dissipative solutions, for thecurve-shortening flow. For given initial conditions, the set of generalizedsolutionsis convex, compact, if not empty. Smooth solutions to thecurve-shortening flow are always unique in this setting.

From conservative to dissipative systemsthrough quadratic change of time, with application to the curve-shortening flow