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A classical approach to the restricted three-body problem is to analyze thedynamics of the massless body in the synodic reference frame. A differentapproach is represented by the perturbative treatment: in particular theaveraged problem of a mean-motion resonance allows to investigate the long-termbehavior of the solutions through a suitable approximation that focuses on aparticular region of the phase space. In this paper, we intend to bridge a gapbetween the two approaches in the specific case of mean-motion resonantdynamics, establish the limit of validity of the averaged problem, and takeadvantage of its results in order to compute trajectories in the synodicreference frame. After the description of each approach, we develop a rigoroustreatment of the averaging process, estimate the size of the transformation andprove that the averaged problem is a suitable approximation of the restrictedthree-body problem as long as the solutions are located outside the Hill'ssphere of the secondary. In such a case, a rigorous theorem of stability overfinite but large timescales can be proven. We establish that a solution of theaveraged problem provides an accurate approximation of the trajectories on thesynodic reference frame within a finite time that depend on the minimaldistance to the Hill's sphere of the secondary. The last part of this work isdevoted to the co-orbital motion (i.e., the dynamics in 1:1 mean-motionresonance) in the circular-planar case. In this case, an interpretation of thesolutions of the averaged problem in the synodic reference frame is detailedand a method that allows to compute co-orbital trajectories is displayed.

Revisiting the averaged problem in the case of mean-motion resonances of  the restricted three-body problem. Global rigorous treatment and application  to the co-orbital motion