Gradient flows, second order gradient systems and convexity
We disclose an interesting connection between the gradient flow of a $\mathcal{C}^2$-smooth function $ψ$ and evanescent orbits of the second order gradient system defined by the square-norm of $\nablaψ$, under adequate convexity assumption. As a consequence, we obtain the following surprising result for two $\mathcal{C}^2$, convex and bounded from below functions $ψ_1$, $ψ_2$: if $||\nablaψ_1||=||\nablaψ_2||$, then $ψ_1=ψ_2 + k$, for some $k\in \mathbb{R}$.
math.OC↗