DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated

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Separable regularized vortex-inspired Hamiltonian comparison system

Definition

For $N \geq 2$ , real weights $Γ_{i}$ , and $ε > 0$ , define the open collision-free configuration set $Q_{c f} : = {q = (q_{1}, \dots, q_{N}) \in (R^{2})^{N} : q_{i} \neq q_{j} for i \neq j}$ . Let $q \in Q_{c f}$ and $p = (p_{1}, \dots, p_{N}) \in (R^{2})^{N}$ . Let $H_{ε} (q, p) = \frac{1}{2} \sum_{i = 1}^{N} ∣ p_{i} ∣^{2} + V_{ε} (q), V_{ε} (q) = \sum_{i < j} Γ_{i} Γ_{j} G_{ε} (q_{i} - q_{j}),$ where $G_{ε}$ is from Smooth mollified Biot-Savart kernel. This is a separable canonical Hamiltonian on $Q_{c f}$ ; its local leapfrog map is the partial composition One-step leapfrog map as force and drift shears restricted as in Local domain restriction for kick, drift, and leapfrog maps. It is a distinct second-order canonical comparison model, not the first-order Biot-Savart vortex system of Equal-probability mean-field and weighted signed vortex particle systems.

Depends on

Used by

Local geometric properties and second-order error of the separable regularized vortex-inspired leapfrog Theorem

Dependency tree · two levels

15 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)
J. Liu and X. Xin, Convergence of Vortex Methods for Weak Solutions to the 2-D Euler Equations with Vortex Sheet Data (standard reference, not scraped)