On the universality of potential well dynamics

Abstract

Given a smooth potential function $V:{\mathbf{R}}^m\to \mathbf{R}$, one can consider the ODE ${\partial }_t^{2}u=-(\nabla V)(u)$ 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 \emph{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 \emph{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 \emph{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 ${\partial }_t^{2}u-\Delta u=-(\nabla V)(u)$ on a torus instead of a particle in a potential well, or if one replaces the target domain ${\mathbf{R}}^m$ by a more general Riemannian manifold.

arXiv comment20 pages, 1 figure. Some typos in the construction of the universal Turing machine encoding fixed