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The restricted three-body problem (R3BP) is an important research area that deals with significant issues in celestial mechanics, such as analyzing asteroid movement behavior and orbit design for space probes. We aim to show the existence of periodic orbits in the planar circular R3BP. To find these orbits, we adapt a variational approach and symmetry.

The restricted three-body problem is an important subject that deals with significant issues referring to scientific fields of celestial mechanics, such as analyzing asteroid movement behavior and orbit designing for space probes. The 2-center problem is its simplified model. The goal of this paper is to show the existence of brake orbits, which means orbits whose velocities are zero at some times, under some particular conditions in the 2-center problem by using variational methods.

The Hill problem models the motion of a particle near a planet. In this paper, we show the existence of symmetric periodic orbits in the spatial Hill problem by using the variational method. We also study the problem under a constraint on a prescribed plane and show the existence of periodic orbits in the problem. The obtained orbits are applicable to artificial satellites around the Earth.