Solving procedure for 3D motions near libration points in CR3BP

Abstract

In a novel approach for solving the equations of the Circular Restricted Three-Body Problem (CR3BP) first formulated in Ershkov (Acta Mech. 228(7):2719–2723, 2017a), we apply in this communication a procedure for solving the Euler-Poisson equations for the momentum equations of the CR3BP near the libration points for uniformly rotating planets having inclined orbits in the solar system with respect to the orbit of the Earth. The system of equations of the CR3BP has been explored with regard to the existence of an analytic way of presentation of the approximated solution in the vicinity of libration points. A new and elegant ansatz is suggested in this publication, whereby, in solving, the momentum equation is reduced to a system of three linear ordinary differential equations of first order in regard to the three components of the velocity of the infinitesimal mass $m$ (dependent on time $t$ ). Under this premise, a proper elegant partial solution has been obtained due to the invariant dependence between temporary components of the solution. We conclude that the system of CR3BP equations does not have the analytical presentation of the solution (in quadratures) even in the vicinity of the libration points except of the generalized Jacobi integral.