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

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Canonical kick and drift shears satisfy the symplectic Jacobian identity

Statement

Let $Q \subseteq R^{d}$ be open, $V \in C^{2} (Q)$ , and $M$ be symmetric positive definite. Put $J = (\begin{array}{cc} 0 & I \\ - I & 0 \end{array})$ . At every point of their open domains, the local kick and drift from Local domain restriction for kick, drift, and leapfrog maps are $C^{1}$ and their Jacobians $A$ satisfy $A^{T} J A = J$ . Here this matrix identity is the finite-regularity meaning of symplecticity; no smooth-manifold definition is being applied to a map of insufficient regularity.

Facts & Assumptions

Given: $Q, V, M$ , the local restrictions of the kick and drift from Local domain restriction for kick, drift, and leapfrog maps, and the canonical matrix $J$ .

[F1]

The published force-kick and drift are $K_{a} (q, p) = (q, p - a \nabla V (q))$ and $D_{b} (q, p) = (q + b M^{- 1} p, p)$ . One-step leapfrog map as force and drift shears

[F2]

The local domain restriction leaves these formulas unchanged wherever the operation is admitted. Local domain restriction for kick, drift, and leapfrog maps

[F3]

Positive definiteness makes $M$ injective, hence invertible in finite dimension; since $M^{T} = M$ , $(M^{- 1})^{T} = (M^{T})^{- 1} = M^{- 1}$ . The Hessian of a $C^{2}$ scalar function is symmetric. Continuous mixed partials of order $k$ are invariant under permutations

Proof

technique · direct block-matrix calculation

1.1F1F2F3givenalgebra

At an admitted point, differentiation of [F1] gives $D K_{a} = (\begin{array}{cc} I & 0 \\ - a \nabla^{2} V (q) & I \end{array}) .$ Writing $C = - a \nabla^{2} V (q) = C^{T}$ , block multiplication gives $D K_{a}^{T} J D K_{a} = (\begin{array}{cc} C - C^{T} & I \\ - I & 0 \end{array}) = J .$

1.2F1F2F3givenalgebra

Differentiation of the drift gives $D D_{b} = (\begin{array}{cc} I & b M^{- 1} \\ 0 & I \end{array}) .$ Writing $B = b M^{- 1} = B^{T}$ , multiplication yields $D D_{b}^{T} J D D_{b} = (\begin{array}{cc} 0 & I \\ - I & B^{T} - B \end{array}) = J .$

2.1step 1.1step 1.2∎

Thus each admitted shear satisfies the claimed identity at every point of its open domain.

Depends on

Used by

Dependency tree · two levels

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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)