Newton’s second law with a semiconvex potential

Abstract

We make the elementary observation that the differential equation associated with Newton’s second law \(m\ddot{\gamma }(t)=-D V(\gamma (t))\) always has a solution for given initial conditions provided that the potential energy V is semiconvex. That is, if \(-D V\) 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.