Gradient flows, second order gradient systems and convexity

Abstract

We disclose an interesting connection between the gradient flow of a ${\mathcal{C}}^2$-smooth function $\psi$ and evanescent orbits of the second order gradient system defined by the square-norm of $\nabla \psi$, under adequate convexity assumption. As a consequence, we obtain the following surprising result for two ${\mathcal{C}}^2$, convex and bounded from below functions ${\psi }_1$, ${\psi }_2$: if $||\nabla {\psi }_1||=||\nabla {\psi }_2||$, then ${\psi }_1={\psi }_2+k$, for some $k\in \mathbb{R}$.