Potential pair forces satisfy Newton's third law

Proved

VirialTheorem.pairForce_antisymm

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

classical-mechanicsvirial-theorem

Let $x_1,\dots,x_N\in\mathbb R^3$ be pairwise distinct and let $V_{jk}:\mathbb R\to\mathbb R$ be pair potentials with $V_{jk}=V_{kj}$ , each $V_{jk}$ ( $j\ne k$ ) differentiable at the distance $\|x_k-x_j\|$ . Then the forces $F_{jk}=-\nabla_{x_k}V_{jk}(\|x_k-x_j\|)$ (with $F_{kk}=0$ ) satisfy

F_{jk}=-F_{kj}\qquad\text{for all } j,k.

Preamble

import Mathlib
import Definitions.Def_virial_theorem_defs

open Filter Topology

Formal statement

namespace VirialTheorem

theorem pairForce_antisymm {N : ℕ} (V : Fin N → Fin N → ℝ → ℝ) (x : Fin N → Space)
    (hsymm : ∀ j k, V j k = V k j)
    (hx : ∀ j k, j ≠ k → x j ≠ x k)
    (hV : ∀ j k, j ≠ k → DifferentiableAt ℝ (V j k) (dist (x k) (x j))) :
    ∀ j k, pairForce V x j k = -pairForce V x k 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_kj = -F_jk)

Read-back

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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 $V_{jk}:\mathbb R\to\mathbb R$ and a configuration $x\in(\mathbb R^3)^N$ . Assume: (i) $V_{jk}=V_{kj}$ as functions for all $j,k$ ; (ii) $x_j\ne x_k$ whenever $j\ne k$ ; (iii) for all $j\ne k$ , $V_{jk}$ is differentiable at $\|x_k-x_j\|$ . Then for all indices $j,k$ (including $j=k$ , where both sides are $0$ ) the pair forces satisfy $F_{jk}(x)=-F_{kj}(x)$ .

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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