Vectorial elements for the Galactic disc tide effects in cometary motion

The classical Matese-Whitman theory of Oort Cloud comet perturbations has been revisited and extended. An explicit solution for the motion of the mean ascending node is given; it involves an elliptic integral of the third kind. Equations of the mean orbit are formulated in terms of the Cartesian components of the Laplace and angular momentum vectors (vectorial elements). The equations are solved in terms of elliptic functions and the solution is free of the ambiguity related to the orientation of the perihelion that was present in previous work. The Cartesian equations of motion for the vectorial elements form a Hamiltonian system of the Lie-Poisson type. This allows them to be integrated numerically by means of Hamiltonian splitting methods. The formulae of such an integrator are derived with a Hamiltonian function split into two parts.