The variational problem for a motion of a spacecraft center of mass in a gravitational field of two fixed centers is considered. The problem of motion in such field was first considered by Euler as an intermediate step towards the solution of the three-body problem. Optimization of motion in the filed of two fixed centers represents a challenging problem due to the integrability issues. Using the cyclic integral, the equations of motion have been shown to be a 12-th order Hamiltonian system of equations. In the absence of general solutions, it is important to study the existence of particular solutions. The relatively unknown method of Dokshevich is used to search for all possible structures of integrals of these equations. This method allows us to determine all possible structures of the complete integrals which have not yet been found for the system of ordinary differential equations, and also it serves as an instrument of finding incomplete integrals. An illustrative example involving the two centers occupied by various pairs of Earth, Moon, Venus and Mars is discussed. Methodology: Dokshevich`s method is used to search for all possible structures of integrals of Hamiltonian system of equations and is used to determine particular integrals. Result: Based on Dokshevich`s method the invariant relations and the particular analytical solutions for intermediate thrust arcs for the problem of minimizing characteristic velocity of point in the case field of two fixed centers.
Volume 12 | 07-Special Issue
Pages: 2126-2137
DOI: 10.5373/JARDCS/V12SP7/20202332