# Newton's second law with a semiconvex potential

## TL;DR

This paper proves that Newton's second law solutions exist when the potential energy is semiconvex, and extends this result to other equations like the Jeans-Vlasov and pressureless Euler equations under similar conditions.

## Contribution

It introduces the concept that semiconvex potentials guarantee solutions for Newton's second law and applies this to various complex dynamical systems.

## Key findings

- Solutions exist for Newton's second law with semiconvex potentials
- Extension of existence results to Jeans-Vlasov and Euler equations
- Verification of solutions under semiconvexity assumptions

## 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.