Radial harmonic functions on a real inner product space: the classification, the Laplacian of radial functions, and the stability of circular orbits for power-law central forces; Ehrenfest's dimensionality argument (1917) as the application
GENERAL THEOREMS FIRST. Throughout, E is a finite-dimensional real inner product space of dimension n, Δ is Mathlib's coordinate-free Laplacian, HarmonicOnNhd its harmonic functions, and f : ℝ → ℝ is twice differentiable on (0, ∞) (two HasDerivAt hypotheses, with derivatives f', f''). (I) THE CLASSIFICATION OF RADIAL HARMONIC FUNCTIONS: (1) for n ≠ 2 and E ≠ 0, the radial function f ∘ ‖·‖ is harmonic on E ∖ {0} if and only if f(r) = a + b r^(2−n) on (0, ∞) for some real a, b; (2) for n = 2, if and only if f(r) = a + b log r — every radial harmonic function with a twice-differentiable profile…
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