Butcher series for Hamiltonian Poisson integrators through symplectic groupoids
Adrien Busnot Laurent and Oscar Cosserat
我们展示了一个新的在辛群胚框架下的预李代数，并由此引入了用于近似任何辛群胚 𝒢⇉ M 上的 Hamilton-Jacobi 解的 Butcher 树的预李形式。这种新的代数方法的意义是双重的。在几何方面，它产生了用于使用 Butcher-Connes-Kreimer Hopf 代数近似 𝒢 的拉格朗日截面的代数运算，并旨在更好地理解 M 的哈密顿微分同胚群。在计算方面，我们定义了一类用于泊松流形上哈密顿动力学的新的泊松积分器。
We exhibit a new pre-Lie algebra in the framework of symplectic groupoids and, in turn, introduce a pre-Lie formalism of Butcher trees for the approximation of Hamilton-Jacobi solutions on any symplectic groupoid 𝒢⇉ M. The impact of this new algebraic approach is twofold. On the geometric side, it yields algebraic operations to approximate Lagrangian bisections of 𝒢 using the Butcher-Connes-Kreimer Hopf algebra and, in turn, aims at a better understanding of the group of Hamiltonian diffeomorp...
