2025/03/06 by Laurent, Adrien Busnot, Cosserat, Oscar
#16T05 #37J39 #41A58 #65L06 #70G45 #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2503.05000
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 \rightrightarrows 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.