Power-law forces: $dG/dt = 2T - n\,V_{\mathrm{TOT}}$

Proved

VirialTheorem.virialG_hasDerivAt_power_law

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

classical-mechanicsvirial-theorem

Throughout, $N$ point particles move in $\mathbb R^3$ ; particle $k$ has constant mass $m_k\in\mathbb R$ , position $r_k(t)$ and velocity $v_k(t)$ , and momentum $p_k=m_kv_k$ . Suppose the only forces are power-law pair forces, $V_{jk}(s)=\alpha_{jk}s^n$ with $\alpha_{jk}=\alpha_{kj}$ , so that at time $t$ each momentum $p_k$ has derivative $F_k(t)=\sum_jF_{jk}(r(t))$ . If moreover $r_k$ has derivative $v_k(t)$ at $t$ and the particles occupy pairwise distinct positions at time $t$ , then

\frac{dG}{dt}(t)=2T(t)-n\,V_{\mathrm{TOT}}(t).

Preamble

import Mathlib
import Definitions.Def_virial_theorem_defs

open Filter Topology

Formal statement

namespace VirialTheorem

theorem virialG_hasDerivAt_power_law {N : ℕ} (m : Fin N → ℝ) (r v : Fin N → ℝ → Space)
    (α : Fin N → Fin N → ℝ) (n : ℝ) (t : ℝ)
    (hα : ∀ j k, α j k = α k j)
    (hr : ∀ k, HasDerivAt (r k) (v k t) t)
    (hp : ∀ k, HasDerivAt (fun s => m k • v k s)
      (netPairForce (powerLawPotential α n) (fun i => r i t) k) t)
    (hx : ∀ j k, j ≠ k → r j t ≠ r k t) :
    HasDerivAt (virialG m r v)
      (2 * kineticEnergy m v t - n * totalPotential (powerLawPotential α n) (fun i => r i t)) t := by sorry

end VirialTheorem

Source

Wikipedia, "Virial theorem" (article supplied as Virial_theorem.pdf), https://en.wikipedia.org/wiki/Virial_theorem, section 'Special case of power-law forces' (dG/dt = 2T - nV_TOT)

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 $N\ge0$ , masses $m_k\in\mathbb R$ , functions $r_k,v_k:\mathbb R\to\mathbb R^3$ , symmetric real coefficients $\alpha_{jk}=\alpha_{kj}$ , a real exponent $n$ and a time $t$ . Assume, at the single time $t$ : each $r_k$ has derivative $v_k(t)$ ; each $s\mapsto m_kv_k(s)$ has derivative equal to the net power-law pair force $F_k$ evaluated at the configuration $(r_0(t),\dots,r_{N-1}(t))$ ; and $r_j(t)\ne r_k(t)$ for $j\ne k$ . Then $G$ has derivative at $t$ equal to

2T(t)-n\sum_k\sum_{j<k}\alpha_{jk}\|r_k(t)-r_j(t)\|^n .

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).

View graph

Power-law forces: dG/dt=2T−n VTOTdG/dt = 2T - n\,V_{\mathrm{TOT}}dG/dt=2T−nVTOT​

Power-law forces: $dG/dt = 2T - n\,V_{\mathrm{TOT}}$