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

Statement

Assume Countable Choice. Fix the particle count $N \geq 2$ , real weights $Γ_{i}$ , and $ε > 0$ . Let $H_{ε} (q, p) = \frac{1}{2} ∣ p ∣^{2} + V_{ε} (q)$ be the separable Hamiltonian of Separable regularized vortex-inspired Hamiltonian comparison system. On an open collision-free configuration set, restrict the leapfrog step to states whose kick and drift stages remain in that set, as in Local domain restriction for kick, drift, and leapfrog maps. Then every admitted step has the following properties:

it is reversible: its inverse on its image is the admitted step with signed step size $- h$ ;
it is symplectic on its open admitted domain;
it preserves local phase volume, meaning Lebesgue measure of every measurable set and its image agree; and
for a fixed $T > 0$ and integer $n \geq 1$ , with $h = T / n > 0$ sufficiently small and exact initialization, if $V_{ε} \in C^{3}$ and the exact path, numerical states, every kick/drift stage, and every line segment required by Local and finite-time trajectory error for symplectic Euler and leapfrog lie in a common compact convex tube $K ⋐ Q_{c f} \times (R^{2})^{N}$ independent of $n$ , with derivatives of the Hamiltonian vector field through order two bounded there, the global trajectory error is $O (h^{2})$ on $[0, T]$ .

Claims 1–3 concern every signed admitted step; claim 4 has the stated positive-step tube hypotheses. The symplecticity and order-two conclusions use the exact hypotheses of the suppliers Local leapfrog and kick-then-drift symplectic Euler are symplectic and Local and finite-time trajectory error for symplectic Euler and leapfrog. This canonical second-order Hamiltonian is a comparison model; it is not the first-order Biot-Savart particle system.

Facts & Assumptions

Given: The smooth separable Hamiltonian, open collision-free domain, admitted local steps, Countable Choice, and (for claim 4) $T > 0$ , $n \geq 1$ , exact initialization, the compact convex tube, and bounded derivatives through order two of the Hamiltonian vector field.

[A1]

Assume Countable Choice for measurable change of variables (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Leapfrog is the symmetric composition of half-kicks and a full drift; the local restriction admits exactly those stages lying in the open configuration set (One-step leapfrog map as force and drift shears, Local domain restriction for kick, drift, and leapfrog maps).

[F2]

Under $V \in C^{2}$ and the admitted-domain hypotheses, local leapfrog is symplectic (Local leapfrog and kick-then-drift symplectic Euler are symplectic).

[F3]

Under $V \in C^{3}$ and the stated compact convex tube, local leapfrog has global order two and its explicit finite-time error estimate (Local and finite-time trajectory error for symplectic Euler and leapfrog).

[F4]

A triangular matrix has determinant equal to the product of its diagonal entries, and determinants multiply (The determinant of a triangular matrix is the product of its diagonal entries, For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ ).

[F5]

Under Countable Choice, a $C^{1}$ diffeomorphism between open sets satisfies change of variables for nonnegative measurable functions (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions).

Proof

1.1F1algebra

Write the kick and drift as $K_{h / 2}$ and $D_{h}$ . Their inverses are $K_{- h / 2}$ and $D_{- h}$ . Since the composition is $K_{h / 2} \circ D_{h} \circ K_{h / 2}$ , its inverse on the image is $K_{- h / 2} \circ D_{- h} \circ K_{- h / 2}$ , exactly the admitted local step with signed size $- h$ . This proves reversibility on the partial-map domains.

1.2givenF2F3

The potential $V_{ε}$ is smooth on the collision-free open set. The admitted local partial map therefore satisfies the hypotheses of [F2], which proves symplecticity. Under the compact convex tube and derivative assumptions in the Given clause, [F3] gives $e_{n} \leq e^{L T} e_{0} + C_{V} T e^{L T} h^{2}$ ; exact initialization has $e_{0} = 0$ , so the finite-time global error is at most $C_{V} T e^{L T} h^{2}$ .

1.3A1F1F4F5algebra

On every admitted open kick domain the kick $K_{a} (q, p) = (q, p - a \nabla V_{ε} (q))$ has derivative $(\begin{array}{cc} I & 0 \\ - a D^{2} V_{ε} (q) & I \end{array})$ . The drift $D_{b} (q, p) = (q + b p, p)$ has derivative $(\begin{array}{cc} I & b I \\ 0 & I \end{array})$ . Both are block triangular with identity diagonal blocks, so each has determinant one by [F4]. The local composition is a $C^{1}$ diffeomorphism from its open admitted domain onto its open image, and multiplicativity [F4] gives determinant one for the composition. Applying [F5] to the map and its inverse yields equality of Lebesgue measures for every measurable set and its image.

2.1F3step 1.1step 1.2step 1.3∎

Steps 1.1, 1.2, and 1.3 prove reversibility, symplecticity and local volume preservation; the bound derived from [F3] proves order two on precisely the stated tube. No energy-conservation or modified-energy conclusion is used.

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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)