Wagner Sessin

Motion of two planets with periods commensurable in the ratio 2:1 solutions of the Hori auxiliary system

It is shown that the Hori auxiliary system for the motion of two planets, whose motions around the Sun have commensurable periods in the ratio 2 : 1, is completely integrable and, an intermediate orbit that includes the effects of the resonance is obtained. The difficulties of classifying some solutions as librations or circulations are discussed.

The generalized canonical form of Hori's method for non-canonical systems

Any dynamical system can be put in generalized canonical form through the introduction of a set of auxiliary ‘conjugate’ variables or momenta and solved by perturbation theory based on Lie series. The application of Horis method for generalized canonical system leads to a new canonical transformation — the Mathieu transformation — defined by the solution of the Hori auxiliary system. This new transformation simplifies the algorithm since the inversion of the solution of the Hori auxiliary system is no longer necessary. In this paper, we wish to show some peculiarities of this technique.

Lagrange variational equations from Hori's method for canonical systems

Hori, in his method for canonical systems, introduces a parameter τ through an auxiliary system of differential equations. The solutions of this system depend on the parameter and constants of integration. In this paper, Lagrange variational equations for the study of the time dependence of this parameter and of these constants are derived. These variational equations determine how the solutions of the auxiliary system will vary when higher order perturbations are considered. A set of Jacobis canonical variables may be associated to the constants and parameter of the auxiliary system that reduces Lagrange variational equations to a canonical form.

Optimal low-thrust limited power transfer between neighbouring quasi-circular orbits of small inclinations around an oblate planet

Abstract A complete analytic study about the influence of Earths oblateness on the optimal low-thrust limited power transfer of small amplitude (orbit correction) between quasi-circular orbits of small inclinations is carried out up to the first order in a small parameter defined by the nondimensional thrust acceleration. The coefficient for the second zonal harmonic J 20 and the nondimensional thrust acceleration are supposed to be the same order of magnitude. Horis method for generalized canonical systems is applied in order to obtain the analytical solution for adjoint and state differential equations. Simple analytic solutions are obtained explicitly for long-time transfer.

A note on the integration of the equations of Lie-Deprit's method for unspecified canonical variables

In this note it will be shown that the equations generated by Lie-Deprits method for unspecified canonical variables could be solved in the same way as Hori did in his method. Here the notations follow those used by Deprit in his paper.

A general algorithm for the determination ofT j (n) andZ j *(n) in Hori's method for non-canonical systems

A general algorithm for the determination ofTj(n) andZj*(n) is deduced. This algorithm is obtained from the general solution of non-homogeneous linear differential equations with variable coefficients in their matricial form. To do this a new functionX*(n) associated withZ*(n) is introduced. Then it is possible to calculateZ*(n) such that it contains secular or mixed secular terms and soT(n) is free from these terms.

The Extended Delaunay Method Applied to First Order Resonance

This paper is concerned with the extended Delaunay method as well as the method of integration of the equations, applied to first order resonance. The equations of the transformation of the extended Delaunay method are analyzed in the (p + 1)/p type resonance in order to build formal, analytical solutions for the resonant problem with more than one degree of freedom. With this it is possible to gain a better insight into the method, opening the possibility for more generalized applications. A first order resonance in the first approximation is carried out, giving a better comprehension of the method, including showing how to eliminate the ‘Poincaré singularity’ in the higher orders.