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Newton’s second law with a semiconvex potential

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  • Ryan Hynd

    (University of Pennsylvania)

Abstract

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

Suggested Citation

  • Ryan Hynd, 2022. "Newton’s second law with a semiconvex potential," Partial Differential Equations and Applications, Springer, vol. 3(1), pages 1-34, February.
    • Handle: RePEc:spr:pardea:v:3:y:2022:i:1:d:10.1007_s42985-021-00136-1
    DOI: 10.1007/s42985-021-00136-1
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    35L65; 60B10; 26B25; 35D30;
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