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Exact RATTLE is symplectic on the constrained phase space

Statement

Let $C \subset M$ be compact and choose a compact phase-space neighborhood $\overline{C}$ of $C$ contained in a regular coordinate chart. There is $h_{0} (\overline{C}) > 0$ such that for $∣ h ∣ < h_{0} (\overline{C})$ the exact map $Ψ_{h}$ of Scaled classical RATTLE step map is defined near $C$ , its image lies in $\overline{C}$ , and its reverse branch is defined near that image; the reverse branch is a local inverse, so $Ψ_{h}$ is a local diffeomorphism near $C$ . On this neighborhood it preserves $ω_{M}$ : $Ψ_{h}^{*} ω_{M} = ω_{M} .$ Here symplecticity of a map means that it is a local diffeomorphism whose pullback preserves the two-form; the form itself is symplectic in the sense of Symplectic form and symplectic manifold.

Facts & Assumptions

Given: The compact phase-space neighborhood in the Statement, the exact selected RATTLE map, and the symplectic cotangent coordinates.

[F1]

Regular holonomic phase space and constrained dynamics: on the regular phase manifold $M$ , $ω_{M}$ is the restriction of the ambient canonical form $ω_{0} = \sum d q_{i} \land d p_{i}$ .

[F2]

The regular holonomic phase space is a symplectic cotangent bundle: in local configuration coordinates $x$ , a consistent ambient momentum represents a cotangent covector $π = D ϕ (x)^{T} p$ and $ω_{M} = \sum d x_{i} \land d π_{i}$ .

[F3]

Scaled classical RATTLE step map: RATTLE uses the two half-force updates, the constrained drift $q_{1} - q = h M^{- 1} p_{1 / 2}$ , and normal constraint impulses at $q$ and $q_{1}$ .

[F4]

Scaled RATTLE equations have unique local multiplier branches: the selected branches define a smooth local map, with a common step range on each compact phase subset.

[F5]

Symplectic form and symplectic manifold: a symplectic form is a smooth closed two-form that is pointwise nondegenerate.

[F6]

Exact RATTLE preserves both constraints and is symmetric: the signed-step map has inverse $Ψ_{- h}$ on the selected regular branches.

Proof

1.1F3givenalgebra

If $h = 0$ , then $Ψ_{0}$ is the identity and preserves $ω_{M}$ . Assume $h \neq 0$ . On a local configuration chart define $L_{h} (q, q_{1}) = \frac{1}{2 h} (q_{1} - q)^{T} M (q_{1} - q) - \frac{h}{2} (V (q) + V (q_{1}))$ . Differentiating gives $- D_{1} L_{h} = p_{1 / 2} + \frac{h}{2} \nabla V (q)$ and $D_{2} L_{h} = p_{1 / 2} - \frac{h}{2} \nabla V (q_{1})$ . By [F3], the first expression differs from $p$ by the normal vector $- h J (q)^{T} λ$ , and the second differs from $p_{1}$ by $h J (q_{1})^{T} μ$ . Each normal vector vanishes on tangent variations of $Q$ , so the restricted cotangent covectors satisfy $p ∣_{T_{q} Q} = - D_{1} L_{h} ∣_{T_{q} Q}$ and $p_{1} ∣_{T_{q_{1}} Q} = D_{2} L_{h} ∣_{T_{q_{1}} Q}$ .

2.1F2step 1.1algebra

Let $θ$ be the canonical one-form on $T^{*} Q$ , $θ_{(q, π)} (δ q, δ π) = π (δ q)$ . The identities in step 1.1 give $Ψ_{h}^{*} θ - θ = d (L_{h} (q, q_{1} (q, p)))$ on the input chart, because both sides applied to an input variation equal $D_{1} L_{h} δ q + D_{2} L_{h} δ q_{1}$ .

3.1F1F2F4F5F6step 2.1∎

Taking the exterior derivative of step 2.1 yields $Ψ_{h}^{*} d θ = d θ$ . By [F1], $ω_{M}$ is the restriction of the ambient canonical form; [F2] identifies it as $- d θ$ in cotangent coordinates. Hence this is $Ψ_{h}^{*} ω_{M} = ω_{M}$ . The branch map is smooth by [F4] and locally invertible by [F6]. By [F2] and [F5], the preserved form is the symplectic form on $M$ ; this proves the stated symplecticity.

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Sources

Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric Numerical Integration, 2nd ed., Chapter VII.1.4, pp. 245–246 (standard reference, not scraped)
Robert I. McLachlan, Klas Modin, Olivier Verdier, and Matt Wilkins, Geometric Generalisations of SHAKE and RATTLE, Foundations of Computational Mathematics 14 (2014), 339–370, §§4.4 and 5.1.2–5.1.3 (standard reference, not scraped)