[Paper Review] On the universality of potential well dynamics

Terence Tao|arXiv (Cornell University)|Jul 8, 2017

Mathematical Dynamics and Fractals9 references3 citations

TL;DR

This paper establishes that the dynamics of a particle in a smooth potential well are universal in the sense that any nonsingular, compact smooth flow can be embedded into such a system if and only if the flow supports a strongly adapted 1-form. Using the Nash embedding theorem, the authors construct a coercive potential well system that is Turing complete, meaning trajectory behavior in the system can encode the halting problem of any Turing machine, rendering certain reachability questions undecidable. The result extends to nonlinear wave equations on a torus and Riemannian manifolds.

ABSTRACT

Given a smooth potential function $V : \mathbf{R}^m o \mathbf{R}$, one can consider the ODE $\partial_t^2 u = -( abla V)(u)$ 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 $\partial_t^2 u - Δu = -( abla V)(u)$ 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.

Motivation & Objective

• To determine whether the dynamics of a particle in a smooth potential well can universally model any smooth, nonsingular, compact flow.
• To identify a geometric obstruction—specifically, the existence of a strongly adapted 1-form—that prevents such embedding for certain flows.
• To construct a potential well system that is Turing complete, encoding the halting problem of any Turing machine within bounded trajectories.
• To extend the universality result from finite-dimensional Euclidean spaces to nonlinear wave equations on a torus and general Riemannian manifolds.
• To demonstrate that undecidability in dynamical systems can arise naturally from smooth potential well dynamics.

Proposed method

• The authors use the Nash embedding theorem to isometrically embed a given compact, nonsingular flow into a Euclidean space with a Riemannian metric.
• They define a strongly adapted 1-form as a necessary and sufficient geometric condition for embedding a smooth flow into potential well dynamics.
• The construction involves embedding the flow into a 4-torus, using disjoint regions for states and tape configurations of a universal Turing machine.
• A diffeomorphism is defined on the torus that simulates the transition rules of a Turing machine via affine maps on disjoint boxes and a skew-product on a Cantor set.
• The system is extended to nonlinear wave equations on a torus by treating the wave equation as a Hamiltonian system on a loop space.
• The key technical innovation is the use of a Cantor set structure in the configuration space to encode infinite tape states and simulate Turing computation.

Experimental results

Research questions

• RQ1 Under what geometric conditions can a smooth, nonsingular, compact flow be embedded into the dynamics of a potential well system?
• RQ2 Is there a universal dynamical system within potential well dynamics that can simulate any first-order ODE on a compact manifold?
• RQ3 Can the halting problem of a Turing machine be encoded in the trajectory behavior of a smooth potential well system?
• RQ4 Does the universality of potential well dynamics extend to infinite-dimensional systems such as nonlinear wave equations on a torus?
• RQ5 What role does the existence of a strongly adapted 1-form play in determining the embeddability of a flow into potential well dynamics?

Key findings

• A smooth, nonsingular, compact flow can be embedded into a potential well system if and only if it supports a strongly adapted 1-form.
• The existence of a strongly adapted 1-form is a necessary and sufficient condition for such an embedding, with a counterexample due to Bryant showing non-embeddability when this form is absent.
• A coercive potential well system can be constructed that is Turing complete, meaning the halting of any Turing machine with a given input is equivalent to a trajectory entering a specific open set.
• There exist bounded trajectories in this system for which it is undecidable whether they enter the target open set, demonstrating intrinsic undecidability in smooth potential well dynamics.
• The universality result extends to nonlinear wave equations on a torus and to potential wells on general Riemannian manifolds, preserving the same embedding and undecidability properties.
• The construction shows that trajectory reachability in such systems can be measured with finite accuracy, implying that undecidability is robust under finite-precision observation.