Trajectory optimization using eccentric longitude formulation

The problem of optimal transfer using a set of nonsingular orbit elements where the eccentric longitude represents the sixth state variable is extended and completed by fully considering the six-state dynamics. Because the eccentric longitude appears in the right-hand sides of the dynamic equations, the use of this particular formulation removes the need for solving the transcendental Kepler equation at each integration step, thereby easing, to some extent, the numerical computations. Furthermore, because the eccentric longitude is being integrated directly, it effectively becomes an independent orbital element such that the adjoint differential equations are derived by assuming that this longitude is independent of the other elements. The variational Hamiltonian is constant throughout the transfer with the boundary conditions given simply in terms of the elements. An example of a general minimum-time transfer using continuous low-thrust is generated duplicating a previous result using the mean longitude formulation to validate the mathematical derivations.