Butcher series for Hamiltonian Poisson integrators through symplectic groupoids

We exhibit a new pre-Lie algebra in the framework of symplectic groupoids and, in turn, use Butcher trees to compute the Taylor series of a solution of a Hamilton-Jacobi equation 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 diffeomorphisms of M . On the computational side, we define a new class of Poisson integrators for Hamiltonian dynamics on Poisson manifolds.