Poisson Midpoint Methods for Nonconstant Structure Matrices

Definitions and mathematical statement

Let $U\subseteq{\mathbb{R}}^d$ be the intended phase domain, $H:U\to{\mathbb{R}}$ the Hamiltonian and $J:U\to{\mathbb{R}}^{d\times d}$ a Poisson structure matrix. In the standard smooth convention, $J^{\mathsf T}=-J$ and \[ \sum_\ell\left(J_{i\ell}\partial_\ell J_{jk} +J_{j\ell}\partial_\ell J_{ki}+J_{k\ell}\partial_\ell J_{ij}\right)=0 \] for all $i,j,k$. These conditions make $\{f,g\}=\nabla f^{\mathsf T}J\nabla g$ a Poisson bracket. The differential equation is \[ y'=J(y)\nabla H(y). \] No invertibility or constant-rank assumption on $J$ is added.

For comparison, the ordinary implicit midpoint step satisfies \[ y_+=y+h\,J\left((y+y_+)/2\right) \nabla H\left((y+y_+)/2\right). \] The source asks whether a nonstandard generalization of this rule exists which is conjugate to a Poisson map for a perturbed tensor. If $\Phi_h$ denotes the sought numerical step and $\chi_h$ an appropriate change of variables, a standard formulation would write \[ \Psi_h=\chi_h^{-1}\circ\Phi_h\circ\chi_h, \] and require a perturbed Poisson tensor $J_h$ satisfying \[ D\Psi_h(y)\,J_h(y)\,D\Psi_h(y)^{\mathsf T} =J_h(\Psi_h(y)). \] The tensor $J_h$ must itself satisfy skew symmetry and Jacobi; an arbitrary perturbed skew matrix is insufficient. This scheme explains the requested conjugacy, but does not choose a method or a perturbation. In particular it does not claim that the ordinary midpoint map has the property for every variable $J$.

The source does not define the admissible generalizations, their order, dependence on $H,J$, allowed evaluations or implicit stages, or whether they must reduce exactly to midpoint for constant $J$. Without such constraints an existence claim for an arbitrary map would not specify the intended numerical construction. Likewise it gives no normalization of $\chi_h$, order or topology of $J_h-J$, step-size range or local/global domain. Whether conjugacy and preservation are exact, hold to a fixed order, or are identities of formal power series is unresolved in the wording. These assertions are distinct and no one is silently chosen.

The literature assessment checked 2026-08-21 describes a general variable-Poisson-tensor question and says special constructions do not supply the requested general conjugacy theorem. Its open label and the verification field's notation-only edit are retained as dated source metadata, not independently checked status. No construction or theorem for a restricted Lie--Poisson case is substituted for the stated general task.

Short English statement

Seek a midpoint-type numerical method for variable Poisson systems that becomes Poisson after a change of variables for a nearby Poisson tensor. The tensor must satisfy Jacobi, and exact, finite-order and formal preservation are different targets. The source leaves the class of methods, perturbation size and conjugacy conventions unspecified.

Sources

• Local UnsolvedMath registry, permanent ID 30000444, source code catalogue entry, "Poisson Midpoint Methods for Nonconstant Structure Matrices," including original and cleaned existence question and verification metadata.
• Geometric Numerical Integration (2006), Oberwolfach Reports, pp. 805--882, DOI 10.4171/owr/2006/14: https://doi.org/10.4171/owr/2006/14.
• Supplied reference "discrete Euler-Poincare and Lie-Poisson equations": https://arxiv.org/abs/math/9909099.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external source has been visited or independently checked here.