Expansion of the planetary Hamiltonian for small eccentricities and inclinations: a vector formalism

We revisit the classical expansion of the planetary Hamiltonian for small eccentricities and mutual inclinations of the orbits, presenting a derivation based entirely on vector formalism. We demonstrate that the secular part of the disturbing function, as well as any other Fourier harmonic, can be expressed in terms of scalar products of vectors lying in the system's invariant plane. Furthermore, we show how to express any such term using the angular momentum and eccentricity vectors of the orbits.