Symplectic structure and applications on Weil bundles

Let M be a smooth manifold and A a Weil algebra. We introduce and discuss the symplectic structure in the Weil bundle \((M^A,\pi ,M)\) $(M^A,\pi ,M)$ and we establish the link between the symplectic structure in M and that in \(M^A.\) $M^A.$ As applications, we discuss Hamiltonian vector fields, symplectic vector fields and poisson structures in both cases.