On the universality of potential well dynamics

Given a smooth potential function $V : \mathbb{R}^m \to \mathbb{R}$, one can consider the ODE $\partial^2_t u = - (\nabla V)(u)$ describing the trajectory of a particle $t \mapsto u(t)$ in the potential well $V$. We consider the question of whether the dynamics of this family of ODE are universal in the sense that they contain (as embedded copies) any first-order ODE $\partial_t u = X(u)$ arising from a smooth vector field $X$ on a manifold $M$. Assuming that $X$ is nonsingular and $M$ is compact, we show (using the Nash embedding theorem) that this is possible precisely when the flow $(M,X)$ supports a geometric structure which we call a strongly adapted $1$-form; many smooth flows do have such a $1$-form, but we give an example (due to Bryant) of a flow which does not, and hence cannot be modeled by the dynamics of a potential well. As one consequence of this embeddability criterion, we construct an example of a (coercive) potential well system which is Turing complete in the sense that the halting of any Turing machine with a given input is equivalent to a certain bounded trajectory in this system entering a certain open set. In particular, this system contains trajectories for which it is undecidable whether that trajectory enters such a set.

Remarkably, the above results also hold if one works instead with the nonlinear wave equation $\partial^2_t u - \Delta u = - (\nabla V)(u)$ on a torus instead of a particle in a potential well, or if one replaces the target domain $\mathbb{R}^m$ by a more general Riemannian manifold.

Keywords

wave maps, potential well, Nash embedding theorem

2010 Mathematics Subject Classification

37C10, 37J99, 74J30

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Received 7 July 2017

Published 8 September 2017