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Lagrangian dynamics and the discovery of cislunar periodic orbits

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Through the art of applying assumptions which restrict the orbital shape and relative inclination of celestial bodies such as the Earth and Moon, trajectory prediction of small bodies of comparatively negligible mass (i.e., a satellite) within a multi-body gravitational system is possible. In this research, Lagrangian analytical methods are applied to formulate succinct derivations of the circular restricted three-body problem (CR3BP), the elliptical restricted three-body problem (ER3BP), and the bicircular restricted four-body problem (BCR4BP). The presence of, or lack thereof, equilibrium points within each dynamical model is discussed and presented in both graphical and tabular form. In terms of application, a form of periodic trajectories within the Earth-Moon system, identified herein as cislunar periodic orbits, is propagated using each of the presented dynamical models. The dynamical variations in these cislunar periodic orbits when transitioning between dynamical models are analyzed and discussed. The methodology behind cislunar periodic orbit generation is also discussed with 33 cislunar periodic orbits presented. Finally, through means of differential correction, it is shown how much error in \(\Delta V\) (\(\Delta e_V\)), the BCR4BP dynamics introduce on the CR3BP solutions for a given number of patchpoints. Results of this analysis show the ER3BP to have a significantly higher perturbative effects than the BCR4BP on cislunar periodic orbits which are closed in the CR3BP. Based on the 33 orbits analyzed, correlation was also observed between the Jacobi constant and the dynamical variations present during the transition of cislunar periodic orbits to higher fidelity models, with larger Jacobi constants being associated with more dynamical variations as an orbit transitions from the CR3BP to both the ER3BP and BCR4BP.

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Data availability

The datasets generated during and/or analyzed during the current study are available following the submittal and approval of a formal institutional request to the Air Force Institute of Technology.


  1. The term “cislunar” refers to the spherical volume of space extending from geosynchronous Earth orbit to and including the Moon’s orbit and the Earth-Moon Lagrange points.

  2. Periodic orbits are defined as orbits which periodically return to their initial conditions (position and velocity) after a constant time of flight or period.

  3. The lunar case is when the satellite is confined to the vicinity of one of the primaries [37]

  4. The comet case is when the satellite is very far from the primaries [37]

  5. The Earth-Moon sphere of influence refers to the spheroidal region about the Earth and Moon within which the Earth and Moon are the primary gravitational influences on a satellite. Past this region, the Sun’s gravitational effect begins to supersede that of the Earth and Moon.

  6. The term “translunar” refers to the volume of space beyond the Moon and its orbit.


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Correspondence to Adam P. Wilmer.

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Wilmer, A.P., Bettinger, R.A. Lagrangian dynamics and the discovery of cislunar periodic orbits. Nonlinear Dyn 111, 155–178 (2023).

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