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Local leapfrog and kick-then-drift symplectic Euler are symplectic

Statement

Under the hypotheses of Canonical kick and drift shears satisfy the symplectic Jacobian identity, every admitted local leapfrog step and every admitted kick-then-drift symplectic Euler step is a $C^{1}$ map with continuous Jacobian satisfying $D Φ^{T} J D Φ = J$ on its open admitted domain. The result concerns symplecticity, not energy conservation.

Facts & Assumptions

Given: $V \in C^{2} (Q)$ , symmetric positive-definite $M$ , the local partial maps and their admitted domains.

[F1]

The local kick, drift, and leapfrog composition have the domains and compositions stated in the local-domain definition. Local domain restriction for kick, drift, and leapfrog maps

[F2]

The symplectic Euler step is the partial composition $D_{h} \circ K_{h}$ . Local kick-then-drift symplectic Euler step

[F3]

Each admitted kick and drift Jacobian satisfies $A^{T} J A = J$ . Canonical kick and drift shears satisfy the symplectic Jacobian identity

[F4]

At each admitted point $z$ , the chain rule gives $D (g \circ f) (z) = D g (f (z)) D f (z)$ , with the outer derivative evaluated at the intermediate point $f (z)$ . The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$

Proof

technique · composition of symplectic shears

1.1givenF1F2

By [F1] and [F2], each step under discussion is a finite composition of admitted local kicks and drifts. The intermediate points lie in $Q$ by the admitted-domain conditions, so every derivative in the composition is defined.

2.1F3F4algebra∎

If $A$ and $B$ satisfy $A^{T} J A = J$ and $B^{T} J B = J$ , then $(A B)^{T} J (A B) = B^{T} (A^{T} J A) B = B^{T} J B = J$ . Applying this identity successively to [F3], with [F4] identifying the Jacobian of each composition, proves the result for both step maps.

Depends on

Used by

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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)