Potential pair forces are central

Proved

VirialTheorem.pairForce_eq_central

by Lucas · Sep 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

classical-mechanicsvirial-theorem

Let $x_1,\dots,x_N\in\mathbb R^3$ and let $V_{jk}:\mathbb R\to\mathbb R$ be pair potentials. Fix $j\ne k$ with $x_j\ne x_k$ , put $r_{jk}=\|x_k-x_j\|>0$ , and assume $V_{jk}$ is differentiable at $r_{jk}$ . Then the force $F_{jk}=-\nabla_{x_k}V_{jk}(\|x_k-x_j\|)$ equals

F_{jk}=-\frac{V_{jk}'(r_{jk})}{r_{jk}}\,(x_k-x_j).

Preamble

import Mathlib
import Definitions.Def_virial_theorem_defs

open Filter Topology

Formal statement

namespace VirialTheorem

theorem pairForce_eq_central {N : ℕ} (V : Fin N → Fin N → ℝ → ℝ) (x : Fin N → Space)
    (j k : Fin N) (hjk : j ≠ k) (hx : x j ≠ x k)
    (hV : DifferentiableAt ℝ (V j k) (dist (x k) (x j))) :
    pairForce V x j k =
      -(deriv (V j k) (dist (x k) (x j)) / dist (x k) (x j)) • (x k - x j) := by sorry

end VirialTheorem

Source

Wikipedia, "Virial theorem" (article supplied as Virial_theorem.pdf), https://en.wikipedia.org/wiki/Virial_theorem, section 'Connection with the potential energy between particles' (F_jk = -grad V_jk)

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) - drafting agent, non-blind

Non-blind read-back. This read-back is not independent testimony. It was written by the same agent that drafted the Lean statements, with full knowledge of the source article and of the intended meaning. Reviewers must not treat it as a blind audit; an independent read-back should still be obtained before relying on it.

Let $N\ge0$ , pair potentials $V_{jk}:\mathbb R\to\mathbb R$ for all ordered pairs, a configuration $x\in(\mathbb R^3)^N$ , and two indices $j\ne k$ with $x_j\ne x_k$ . Write $d=\|x_k-x_j\|$ (so $d>0$ ). Assume $V_{jk}$ is differentiable at $d$ . Then the pair force (defined as minus the gradient at $x_k$ of $y\mapsto V_{jk}(\|y-x_j\|)$ ) equals

-\frac{V_{jk}'(d)}{d}\,(x_k-x_j),

where $V_{jk}'(d)$ is the ordinary derivative of $V_{jk}$ at $d$ .

Human review

Endorsed by Shuze Chen · Oct 8, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 8, 2026
Confirmed by the mission captain (proposal self-audit).

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