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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Oscillator Verlet: a preserved quadratic form and bounded original-energy oscillation

Example

Take $d = 1$ , $U = R^{2}$ , $Π = R$ , $M = (1)$ , and $V (q) = q^{2} / 2$ in The splitting Hamiltonian $H = A + B$ , the drift and kick maps, and the velocity-Verlet map. Thus the dimensionless oscillator has $H (q, p) = (q^{2} + p^{2}) / 2$ . For each initial state $z_{0} = (q_{0}, p_{0}) \in U$ , velocity Verlet is the matrix map $z \mapsto S_{h} z$ , where

$S_{h} = (\begin{array}{cc} 1 - h^{2} / 2 & h \\ - h (1 - h^{2} / 4) & 1 - h^{2} / 2 \end{array}) .$

For $0 < h < 2$ , put $α = 1 - h^{2} / 4$ and $I_{h} (q, p) = (α q^{2} + p^{2}) / 2$ . This positive quadratic form is exactly preserved. Every iterate satisfies

$∣ H (z_{n}) - H (z_{0}) ∣ \leq \frac{h^{2}}{4 α} I_{h} (z_{0}) .$

The conclusion is an explicit exact-arithmetic all-step fact for this linear example. At $h = 2$ positivity fails; at $h > 2$ the map has a real eigenvalue of modulus greater than one.

Verification

Given: The domain, mass matrix, potential, arbitrary initial state $z_{0} \in U$ , and the velocity-Verlet kick-drift-kick formula from the splitting definition; $0 < h < 2$ .

[F1] The splitting Hamiltonian $H = A + B$ , the drift and kick maps, and the velocity-Verlet map gives $K_{h / 2} (q, p) = (q, p - h q / 2)$ and $D_{h} (q, p) = (q + h p, p)$ ; The drift, kick and velocity-Verlet maps are canonical: $(D C)^{T} J D C = J$ and $C^{*} ω = ω$ proves exact symplecticity.

1.1givenF1algebra

Matrix and invariant. Multiplying the three displayed shear matrices gives $S_{h}$ as stated. Direct multiplication gives $S_{h}^{T} diag (α, 1) S_{h} = diag (α, 1)$ , using $(1 - h^{2} / 2)^{2} + h^{2} α = 1$ . Thus $I_{h} (z_{n}) = I_{h} (z_{0})$ and $q_{j}^{2} \leq 2 I_{h} (z_{0}) / α$ .

2.1step 1.1algebra∎

Original energy and boundary. Since $H - I_{h} = h^{2} q^{2} / 8$ , $∣ H (z_{n}) - H (z_{0}) ∣ = (h^{2} / 8) ∣ q_{n}^{2} - q_{0}^{2} ∣ \leq h^{2} I_{h} (z_{0}) / (4 α)$ . At $h = 2$ the invariant matrix is singular. For $h > 2$ , $\det S_{h} = 1$ and $tr S_{h} = 2 - h^{2} < - 2$ , so the real reciprocal eigenvalues have one modulus greater than one. This explains the step restriction without asserting stability at its excluded endpoint.

Depends on

Dependency tree · 2 levels

Sources

Hairer, Geometric Numerical Integration, Lecture 3: backward error analysis, Sections 1–3 and 5; local analytic estimates supplied here