The Conservation of Area Integrals in Averaging Transformations

It is shown for the two-planetary version of the weakly perturbed two-body problem that, in a system defined by a finite part of a Poisson expansion of the averaged Hamiltonian, only one of the three components of the area vector is conserved, corresponding to the longitudes measuring plane. The variability of the other two components is demonstrated in two ways. The first is based on calculating the Poisson bracket of the averaged Hamiltonian and the components of the area vector written in closed form. In the second, an echeloned Poisson series processor (EPSP) is used when calculating the Poisson bracket. The averaged Hamiltonian is taken with accuracy to second order in the small parameter of the problem, and the components of the area vector are expanded in a Poisson series.