Approximating Symplectic Realizations: A General Framework for the Construction of Poisson Integrators

TL;DR

The paper presents a general framework for Poisson integrators on arbitrary Poisson manifolds by decoupling geometric approximation via symplectic realization data from dynamic approximation of the flow. It establishes that approximate realization data induce Poisson diffeomorphisms preserving the Poisson tensor and Casimirs to order $n$, and that combining this geometry with a dynamic approximation yields a complete integrator of order $\min\{n,m\}$ for both geometry and dynamics. Two concrete dynamic strategies are developed: a Hamilton-Jacobi–based approach (K-HJ) and a collective-integrator approach (K-Collective), each with formal order guarantees and practical implementation details. The authors demonstrate the methods on several examples, including $\mathfrak{so}^*(3)$, Lotka-Volterra dynamics, and noncanonical symplectic structures, showing substantial preservation of Casimirs, Hamiltonians, and the Poisson tensor. This framework offers a practical route to geometric Poisson integration without requiring explicit leaf parametrizations, with potential extensions to learning Poisson dynamics and higher-order schemes.

Abstract

While the construction of symplectic integrators for Hamiltonian dynamics is well understood, an analogous general theory for Poisson integrators is still lacking. The main challenge lies in overcoming the singular and non-linear geometric behavior of Poisson structures, such as the presence of symplectic leaves with varying dimensions. In this paper, we propose a general approach for the construction of geometric integrators on any Poisson manifold based on independent geometric and dynamic sources of approximation. The novel geometric approximation is obtained by adapting structural results about symplectic realizations of general Poisson manifolds. We also provide an error analysis for the resulting methods and illustrative applications.

Figures (10)

• Figure 1: Illustration of the error of the symplectic realization at the point $(2, 3, 3, 1, 2, 3)$. ( Top) As explained in the beginning of this section, we consider an order $n=10$ approximation $\hat{\alpha}_\epsilon$ to $\alpha_\epsilon$ computed using Mathematica. We showcase the error, measured as the square of the differences in the realization, following \ref{['sym:error']}. We observe a small error even for large values of $\epsilon$. ( Middle) The same plot but restricted to the interval $[0,0.8]$, where the error shows the biggest increment. We see that up to $\epsilon = 0.6$ the error is quite small. ( Bottom) Representation of the logarithm of the error versus the logarithm of epsilon. We observe numerically a slope of approximately $10.99$ for small values of $\epsilon$. Taking into account that derivatives have been computed numerically, which introduces an error, this matches our theoretical prediction of the error being $\mathcal{O}(\epsilon^{n +1}) = \mathcal{O}(\epsilon^{11})$.
• Figure 2: ( Top) We consider an order $4$ approximation to $\alpha_\epsilon$. We observe a small error even for quite large values of $\epsilon$. ( Middle) The same plot as in the figure above, but with $\epsilon$ ranging on the interval $[0,0.2]$. ( Bottom) Plot of the figure above, but representing the the logarithm of the error versus the logarithm of epsilon. We observe numerically a slope of approximately $4.9$ for small values of $\epsilon$. Taking into account that derivatives have been computed numerically, this matches our theoretical prediction of the error being $\mathcal{O}(\epsilon^{n +1}) = \mathcal{O}(\epsilon^{5})$.
• Figure 3: ( Top) Evolution of the difference of the Casimir (value at iteration - initial value). We observe conservation of the Casimir of order higher than expected theoretically. This is probably due to the fact that the Poisson structure is linear. ( Bottom) Evolution of the difference of the Hamiltonian (value at current iteration - initial value) through the simulated trajectory. Due to the low order approximation of the dynamics we observe oscillations of the expected order. These plots clearly illustrate how dynamics and geometry can be approximated to different orders.
• Figure 4: Evolution of Casimir and Hamiltonian difference throughout a trajectory of ODE45. In this case, dynamics and geometry are conserved at a similar order (order $5$).
• Figure 5: ( Top) Plot of the tensor conservation error as a function of epsilon. We observe a very small error even for large values of $\epsilon$. ( Bottom) Plot of the same data as in the figure above, but representing the logarithm of $\epsilon$ versus the logarithm of the error. We observe a slope even larger than the expected one. This is probably due to the fact that the Poisson structure is linear and henceforth easy to approximate.

Theorems & Definitions (30)

• Example 1: Dual of a Lie Algebra, $\mathfrak{g}^*$
• Example 2: Canonical Symplectic Form in the Cotangent Bundle
• Example 3: Type I Generating Functions
• Definition 1: Symplectic Realization
• Definition 2
• Remark 1: Dual pairs
• Remark 2: Symplectic groupoids
• Definition 3: Lagrangian Bisection and the induced mapping
• Theorem 4
• Remark 3: Rescaling $\pi$