Newton's second law with a semiconvex potential

We make the elementary observation that the differential equation associated with Newton's second law m gamma<spacing diaeresis>(t)=-DV(gamma(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ddot{\gamma }(t)=-D V(\gamma (t))$$\end{document} always has a solution for given initial conditions provided that the potential energy V is semiconvex. That is, if -DV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-D V$$\end{document} satisfies a one-sided Lipschitz condition. We will then build upon this idea to verify the existence of solutions for the Jeans-Vlasov equation, the pressureless Euler equations in one spatial dimension, and the equations of elastodynamics under appropriate semiconvexity assumptions.