Lagrange's identity: $\tfrac12 \ddot I = 2T + V_{\mathrm{TOT}}$ for $n=-1$

Proved

VirialTheorem.lagrange_identity

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 each $r_k$ is differentiable at all times with derivative $v_k$ , the particles are pairwise distinct at time $t$ , and at time $t$ each momentum has derivative equal to the net force from inverse-distance pair potentials $V_{jk}(s)=\alpha_{jk}s^{-1}$ with $\alpha_{jk}=\alpha_{kj}$ . Then

\frac12\frac{d^2I}{dt^2}(t)=2T(t)+V_{\mathrm{TOT}}(t).

Preamble

import Mathlib
import Definitions.Def_virial_theorem_defs

open Filter Topology

Formal statement

namespace VirialTheorem

theorem lagrange_identity {N : ℕ} (m : Fin N → ℝ) (r v : Fin N → ℝ → Space)
    (α : Fin N → Fin N → ℝ) (t : ℝ)
    (hα : ∀ j k, α j k = α k j)
    (hr : ∀ k s, HasDerivAt (r k) (v k s) s)
    (hp : ∀ k, HasDerivAt (fun s => m k • v k s)
      (netPairForce (powerLawPotential α (-1)) (fun i => r i t) k) t)
    (hx : ∀ j k, j ≠ k → r j t ≠ r k t) :
    (1 / 2 : ℝ) * deriv (deriv (momentOfInertia m r)) t =
      2 * kineticEnergy m v t + totalPotential (powerLawPotential α (-1)) (fun i => r i 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' (Lagrange's identity)

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 $\alpha_{jk}=\alpha_{kj}$ and a time $t$ . Assume: for every $k$ and every time $s$ , $r_k$ has derivative $v_k(s)$ at $s$ ; at the single time $t$ , each $s\mapsto m_kv_k(s)$ has derivative equal to the net pair force for potentials $V_{jk}(s)=\alpha_{jk}s^{-1}$ at the configuration $r(t)$ ; and $r_j(t)\ne r_k(t)$ for $j\ne k$ . Then, writing $\ddot I(t)$ for the derivative at $t$ of the derivative function of $I(s)=\sum_km_k\|r_k(s)\|^2$ (Lean's deriv, which is $0$ at points of non-differentiability),

\tfrac12\,\ddot I(t)=2T(t)+\sum_k\sum_{j<k}\alpha_{jk}\|r_k(t)-r_j(t)\|^{-1}.

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

Lagrange's identity: 12I¨=2T+VTOT\tfrac12 \ddot I = 2T + V_{\mathrm{TOT}}21​I¨=2T+VTOT​ for n=−1n=-1n=−1

Lagrange's identity: $\tfrac12 \ddot I = 2T + V_{\mathrm{TOT}}$ for $n=-1$