We use local symplectic Lie groupoids to approximate Hamiltonian dynamics for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a neighborhood of the unit manifold, that, in turn, provide the desired approximation. This approximation provides in fact a new tool in numerical analysis for finite-dimensional conservative mechanics. I will finish this talk with a few perspectives in Hamiltonian dynamics, for instance, an extension to Hamiltonian PDEs.