AlgorithmAlgorithm: AI-adaptedProof: Not applicablePipeline-generatedjudge 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.

Local kick-then-drift symplectic Euler step

Definition

Input an admitted state $(q, p) \in Q \times R^{d}$ , step $h > 0$ , $V \in C^{1} (Q)$ , and symmetric positive-definite $M$ . First set $p^{+} = p - h \nabla V (q)$ . Then set $q^{+} = q + h M^{- 1} p^{+}$ . If $q^{+} \notin Q$ , return domain-failure; otherwise return $(q^{+}, p^{+})$ . This fixes the kick-then-drift ordering. The returned map is the partial composition $D_{h} \circ K_{h}$ in the notation of Local domain restriction for kick, drift, and leapfrog maps.

For exact mathematical statements the displayed operations are evaluated exactly. A finite implementation that cannot certify the state and force evaluation returns inconclusive; it does not report a step outside $Q$ as successful.

Depends on

Local domain restriction for kick, drift, and leapfrog maps

Used by

Dependency tree · two levels

2 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

Ernst Hairer, Geometric Numerical Integration, TU München Lecture 2: Symplectic integrators (standard reference, not scraped)
Nikolaj Nordkvist and Poul G. Hjorth, Classical Mechanics and Symplectic Integration (standard reference, not scraped)