Formal solution of the asteroid motion near the 2:1 commensurability

Abstract

In a previous paper (Ammar in Proc. Math. Phys. 77:99, 2002) the statement of the problem was formulated and the basic equations of motion were formed in terms of variables suitable for the applications in the problem of asteroid motion close to 2:1 commensurability. The short period terms has been eliminated up to the first order in masses O(μ), using a perturbation approach based on the Lie series, the problem is reduced to that of secular resonance one. In the present work the extended Delaunay method has been applied to develop the Hamiltonian and the generator as a power series in $\sqrt{\varepsilon}$ rather than the power of e, where e is a small parameter of order of the relative mass of the perturber. Hamilton–Jacobi method were used as a method of integration of the equations of the dynamical system in order to build a formal solution for the resonant problem of the type 2:1 with one degree of freedom.