DefinitionDefinition: 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.

Matched variational Hamiltonian branch for periodic PIC

Definition

Use the smooth assignment branch of Periodic PIC deposition shape and gather operators, real particle charges $q_{p}$ , positive masses $m_{p}$ , and its neutralized grid density $ρ (X)$ . Let $ϕ (X) = A_{h}^{†} ρ (X)$ denote the unique mean-zero solution from Periodic centered-difference Poisson solvability and zero-mean gauge. Define the grid potential energy $U_{h} (X) = \frac{1}{2} ⟨ ρ (X), A_{h}^{†} ρ (X) ⟩_{h} = \frac{1}{2} ⟨ ρ (X), ϕ (X) ⟩_{h}$ and the finite-dimensional separable Hamiltonian $H_{h} (X, p) = \frac{1}{2} p^{T} M^{- 1} p + U_{h} (X), M = diag (m_{1} I_{d}, \dots, m_{P} I_{d}) .$ The corresponding canonical ODE is $\dot{X} = M^{- 1} p$ , $\dot{p} = - \nabla_{X} U_{h} (X)$ . Its force is the variational force obtained by differentiating the deposited shape in $U_{h}$ ; this definition does not identify that force with ordinary field interpolation $G_{X} E$ .

Depends on

Used by

Dependency tree · two levels

7 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

R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, Section 5-5 (standard reference, not scraped)
Nikolaj Nordkvist and Poul G. Hjorth, Classical Mechanics and Symplectic Integration, §3.5 (standard reference, not scraped)