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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^n$ be an open set, $n \\ge 2$, and let $\\varphi \\colon U \\to \\R$ be a [[harmonic function|0DEB0JMN0W]].\nThen $\\varphi$ has no strict local minimum and no strict local maximum in $U$.\nConsequently, consider a charged particle with charge $q$ that moves under the force $F = -q \\nabla \\varphi$.\nNo [[equilibrium point|0N8NGKDZDZ]] in $U$ is a strict local minimum of its potential energy $q\\varphi$.\nIn this sense the particle has no stable equilibrium in $U$.\n\n## Remarks\n\nProof: suppose that $\\varphi(x) \u003e \\varphi(a)$ for all $x \\ne a$ in a ball $B(a, \\rho) \\subseteq U$.\nTake $0 \u003c r \u003c \\rho$.\nBy the [[mean value property|0HYPR64JM2]], $\\varphi(a)$ is the average of $\\varphi$ over the sphere of radius $r$ about $a$.\nBut every value on that sphere is larger than $\\varphi(a)$, a contradiction.\nA strict local maximum is excluded in the same way, or by applying the argument to $-\\varphi$.\nIn physics, $\\varphi$ is the potential of charges that lie outside $U$; it is harmonic in the charge-free region $U$.\nExample: in $\\R^3$, with $e_1 = (1, 0, 0)$ and $e_2 = (0, 1, 0)$, let $\\varphi(x) = 1/\\abs{x - e_1} + 1/\\abs{x + e_1}$ be the potential of two unit charges at $\\pm e_1$.\nThe origin is an equilibrium point by symmetry.\nAlong the $x_1$-axis, $\\varphi(t e_1) = 2/(1 - t^2)$ has a minimum at $t = 0$, but along the $x_2$-axis, $\\varphi(t e_2) = 2/\\sqrt{1 + t^2}$ has a maximum, so the origin is a saddle point.\nThe theorem is about potentials that are harmonic; diamagnetic materials, feedback control and time-varying fields are classical ways around it in physics.\n\n## Explanations\n\n### L1 human\n\nYou cannot hold a charged particle at rest in a stable way using fixed charges alone.\nAt any point where the forces balance, there is a direction in which a small push makes the particle run away.\nMathematically, a harmonic function cannot have a bowl-shaped minimum, because its value at a point is the average of its values around the point.\n\n### L2 human\n\nThe theorem is a direct consequence of the mean value property, and it is a weak form of the strong maximum and minimum principles.\nIt applies to every inverse-square force law, so it also rules out stable static configurations of point masses under Newtonian gravity.\nAt an equilibrium of a harmonic potential, the Hessian has trace $0$, so it cannot be positive definite.\nThis is another way to see that an equilibrium cannot be a nondegenerate minimum.\n",
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