  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.
