Symmetric Periodic Solutions in the Generalized Sitnikov Problem with Homotopy Methods

The paper investigates a generalization of the classical Sitnikov problem, concentrating on the movement of a satellite along the [Formula: see text]-axis as it interacts with [Formula: see text] primary bodies in periodic motion. It establishes the existence of an infinite number of even and antiperiodic solutions with increasing periods. The proof employs the Leray–Schauder degree theory to trace the critical points of action functionals, using a homotopy from solutions when the primary bodies are transformed into circular orbits.