TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Leapfrog geometry and finite-time order for the matched periodic PIC Hamiltonian

Statement

Let $H_{h} (X, p) = \frac{1}{2} p^{T} M^{- 1} p + U_{h} (X)$ be Matched variational Hamiltonian branch for periodic PIC, with $M$ symmetric positive definite and $U_{h} \in C^{3} (R^{d P})$ periodic. Let $Φ_{Δ t} = K_{Δ t / 2} \circ D_{Δ t} \circ K_{Δ t / 2}$ be the leapfrog map with the kick and drift conventions of One-step leapfrog map as force and drift shears. Then:

For every signed step $Δ t$ , $Φ_{Δ t}^{- 1} = Φ_{- Δ t}$ .
Every step is symplectic.
Assuming Countable Choice, every step preserves Lebesgue volume on phase space.
On a finite interval $[0, T]$ , if the exact and numerical paths, all kick/drift stages, and all line segments in the comparison lie in a common compact convex tube and the Hamiltonian vector field has bounded derivatives through order two there, the leapfrog trajectory error is at most $C_{V} T e^{L T} Δ t^{2}$ for exact initialization and sufficiently small $Δ t = T / n$ .

The exact continuous flow conserves $H_{h}$ by The matched periodic PIC grid force is a gradient and its ODE conserves energy. No claim says the leapfrog map preserves $H_{h}$ exactly or that modified-energy estimates hold outside the analytic tube hypotheses of the geometric-integration suppliers.

Facts & Assumptions

Given: A smooth periodic potential, symmetric positive-definite mass matrix, the kick/drift convention, an admitted step, and (for the finite-time error) the stated compact convex tube and exact initialization.

[A1]

Countable Choice is assumed only in claim 3 (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

One-step leapfrog map as force and drift shears: $K_{a} (q, p) = (q, p - a \nabla U_{h} (q))$ , $D_{b} (q, p) = (q + b M^{- 1} p, p)$ , and $Φ_{Δ t} = K_{Δ t / 2} \circ D_{Δ t} \circ K_{Δ t / 2}$ .

[F2]

Canonical kick and drift shears satisfy the symplectic Jacobian identity: for $U_{h} \in C^{2}$ and $M$ symmetric positive definite, each kick and drift Jacobian satisfies $A^{T} J A = J$ .

[F3]

Local leapfrog and kick-then-drift symplectic Euler are symplectic: the admitted local leapfrog composition is $C^{1}$ and symplectic under those hypotheses.

[F4]

Local and finite-time trajectory error for symplectic Euler and leapfrog: on a common compact convex tube with $U_{h} \in C^{3}$ , bounded derivatives through order two of the Hamiltonian vector field, exact initialization, and admitted stages, leapfrog has finite-time global error at most $C_{V} T e^{L T} Δ t^{2}$ .

[F5]

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions: under Countable Choice a $C^{1}$ diffeomorphism changes Lebesgue measure by the absolute value of its Jacobian determinant.

[F6]

The matched periodic PIC grid force is a gradient and its ODE conserves energy: the continuous semidiscrete Hamiltonian flow conserves the exact $H_{h}$ .

Proof

1.1F1algebra

The kicks and drifts in [F1] have inverses $K_{- a}$ and $D_{- b}$ . Reversing the composition gives $Φ_{Δ t}^{- 1} = K_{- Δ t / 2} \circ D_{- Δ t} \circ K_{- Δ t / 2} = Φ_{- Δ t}$ . All maps are global on the periodic lift, so no intermediate-domain restriction is lost.

1.2F1F2F3given

The potential is smooth, hence $C^{2}$ , and $M$ is symmetric positive definite. Each admitted kick and drift satisfies the symplectic Jacobian identity [F2]; composing the three maps preserves that identity by the chain rule, as stated in [F3].

1.3A1F1F5givenalgebra

The Jacobian of a kick is $(\begin{array}{cc} I & 0 \\ - \frac{Δ t}{2} \nabla^{2} U_{h} & I \end{array})$ and the Jacobian of a drift is $(\begin{array}{cc} I & Δ t M^{- 1} \\ 0 & I \end{array})$ . Each is block triangular with identity diagonal blocks, so its determinant is $1$ ; multiplicativity gives $\det D Φ_{Δ t} = 1$ . The global kick/drift maps are $C^{1}$ diffeomorphisms, so [A1] permits applying [F5] to the map and its inverse, proving preservation of Lebesgue volume.

2.1F4F6given∎

Under the tube and smoothness hypotheses, [F4] gives the stated global order-two error bound. The continuous invariant is the separate identity [F6]; it is not an assertion about the discrete map.

Depends on

Used by

Leapfrog changes the energy of an exact harmonic test mode Example

Dependency tree · two levels

41 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nikolaj Nordkvist and Poul G. Hjorth, Classical Mechanics and Symplectic Integration (standard reference, not scraped)
R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, Chapter 4 (standard reference, not scraped)