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 illus-trates some symmetry properties that hold for optimal space trajectories and can considerably simplify their determination. The existence of symmetry prop-erties 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 rendez-vous 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 dura-tion 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, estab-lishes 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