We introduce a pre-Lie formalism of Butcher trees for the approximation of Hamilton-Jacobi solutions on any symplectic groupoid $\mathcal{G} \rightrightarrows M.$ The impact of this new algebraic approach is twofold. On the geometric side, it yields algebraic operations to approximate Lagrangian bisections of $\mathcal{G}$ using the Butcher-Connes-Kreimer Hopf algebra and, in turn, aims at a better understanding of the group of Poisson diffeomorphisms of $M.$ On the computational side, we define a new class of Poisson integrators for Hamiltonian dynamics on Poisson manifolds.

翻译：本文引入了一种基于布彻树的预李形式体系，用于在任意辛群胚 $\mathcal{G} \rightrightarrows M$ 上近似求解哈密顿-雅可比方程。这一新的代数方法具有双重影响。在几何层面，它利用布彻-康内斯-克赖默 Hopf 代数，产生了一系列代数运算来近似 $\mathcal{G}$ 的拉格朗日双截面，进而旨在深化对 $M$ 上泊松微分同胚群的理解。在计算层面，我们定义了一类新的泊松积分器，用于求解泊松流形上的哈密顿动力学方程。