Newtonian Dynamics in Metric Spaces

Abstract

We develop a variational framework for second-order, frictionless (Newtonian) dynamics in metric spaces, extending the metric theory of curves of maximal slope from gradient flows to evolutions with inertia. The construction rests on three pillars, mirroring the first-order theory. First, we introduce a class of second-order absolutely continuous curves, characterized by a geometric--variational inequality on families of tangent geodesics, and define a metric notion of acceleration. Second, we define curves of Newtonian motion as the curves achieving equality in an energy inequality involving an upper gradient--Hessian pair, which in Hilbert spaces is equivalent to Newton's equation $\ddotγ= -\nabla E(γ)$. Third, we propose a two-step predictor--corrector minimizing movement scheme and show that its discrete interpolants satisfy approximate versions of these inequalities. Under extendability and interceptor conditions on geodesics---which are satisfied, for instance, by geodesically uniformly complete, locally compact spaces with curvature bounded below---and under suitable regularity assumptions on the energy, we prove that the scheme converges, up to subsequences, to a curve of Newtonian motion. As an illustration, we recover the linear wave equation as a curve of Newtonian motion driven by the Dirichlet energy. This paper is complemented by the companion paper [Fornasier, Klemenc, Scagliotti, "A Lions Lift Framework for Gradient and Newtonian Evolutions in Wasserstein spaces"], which specifies and adapts the results to the Wasserstein metric space $(\mathcal{P}_2,W_2)$ through Lions lift, which explains Vlasov equations as Newtonian flows on spaces of probability measures.