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Proposition

Suppose $U\colon\Omega\to\mathbb R$ is a non-constant harmonic function, i.e. $U\in\mathcal C^\omega$, i.e. analytic, and $\Delta U=0$, where $\Omega\subseteq\mathbb R^n$ is a region. Then every equilibrium of the system $\ddot x=-\nabla U$ is unstable in the sense of Lyapunov.

(In other words, suppose $(x_0,y_0)$ is an equilibrium of $\dot x=y,\dot y=-\nabla U$, then $(x_0,y_0)$ is unstable.)

Discussion

In electrostatic, the preceding statement is known as Earnshaw's theorem. However, the proofs I saw aren't completely rigorous. They rely on the maximum principle of harmonic functions, and the following statement:

If $\nabla U(x_0)=0$ and $x_0$ isn't a strict local minimum, then the equilibrium $x=x_0,\dot x=0$ is unstable.

However, in V.I.Arnold's Mathematical Methods of Classical Mechanics, it's said that

It seems likely that in an analytic system with $n$ degrees of freedom, an equilibrium position which is not a minimum point (of the potential energy) is unstable; but this has never been proved for $n>2$.

In addition, $\Delta U=0$ implies that the Hessian $H$ of $U$ at $x_0$ satisfies $\operatorname{tr}H=0$. If $H$ has a negative eigenvalue (otherwise $H=0$), then by spectrum theorem of symmetric matrix $H$ and the theory of linearization, system is unstable at $x_0$. However, there's no evident that $H\neq0$.

Any idea? Thanks!

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Here is a general and rigorous argument for Earnshaw's Theorem using a probabilistic definition of unstable: namely, with probability one any perturbed system moves away under the gradient flow. We assume the potential function is harmonic which the electrostatic potential is in dimension three. Here is the Argument: The gradient preserves volume [div grad = laplacian]. Consider the compact region between two different levels of the potential.Then the volume mass must flow in from the higher level and out at the lower level with only a set of measure zero being exceptional. This is easy to make rigorous using the poincare recurrence idea that the alternative to the latter statement would create a recurrent piece of the dynamics which contradicts the gradient nature of the flow QED.

I am thinking Arnold's point about real analytic potentials meant the following: He was considering the dynamics near an isolated critical point to be understood from the point of view of normal forms: being analytic means these critical points are isolated and from his and others' research there should be a meaningful and interesting set of algebraic normal forms. These, apparently were understood completely in dimension two but have not been understood completely in higher dimensions.

I am also thinking the physical motivation behind Earnshaw's Theorem is consistent with the coarser probabilistic version of unstable mentioned above.

Dennis Sullivan

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  • $\begingroup$ I much appreciated this insight! $\endgroup$ – Chris Gerig Oct 5 '17 at 5:43

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