Gradient flows, second-order gradient systems and convexity

We disclose an interesting connection between the gradient flow of a C 2 - smooth function ψ and evanescent orbits of the second order gradient system defined by the square-norm of ∇ψ, under adequate convexity assumption. As a consequence, we obtain the following surprising result for two C 2 , convex and bounded from below functions ψ1, ψ2: if ||∇ψ1|| = ||∇ψ2||, then ψ1 = ψ2 + k, for some k ∈