Symmetry Properties of Optimal Space Trajectories

The determination of minimum-fuel or minimum-time space trajectories has been pursued for decades, using different methods of solution. This work illustrates some symmetry properties that hold for optimal space trajectories and can considerably simplify their determination. The existence of symmetry properties is demonstrated in the presence of certain boundary conditions for the problem of interest. If the linear Euler-Hill equations of motion are used, a pair of properties are proven to hold for minimum-time and minimum-fuel rendezvous trajectories, both in two and in three dimensions. With regard to minimum-fuel paths, the primer vector theory predicts the existence of several powered phases, divided by coast arcs. In general, the optimal thrust sequence and duration depend on the time evolution of the switching function, and can be proven to satisfy interesting symmetry properties, provided that a pair of elementary boundary conditions holds. In the context of the circular restricted three-body problem, the theorem of image trajectories, proven about 50 years ago, establishes the symmetry properties of feasible trajectories. In this work this theorem is extended to optimal trajectories, thus substantiating a conjecture that dates back to the 60's.