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Essays

Vol. 158 No. 2 (2025): Rendiconti. Mathematics and Natural Sciences Class

Collinear homographic orbits in the general problem of N bodies

  • Antonio Giorgilli
  • Ugo Locatelli
  • Marco Sansottera
DOI
https://doi.org/10.3280/rndoa2025oa21163
Submitted
ottobre 1, 2025
Published
2026-02-17

Abstract

We revisit the problem of collinear solutions of the problem of N bodies, as investigated by Euler and Lagrange. Unlike most existing studies, we consider a general class of attractive forces. In this respect, we follow Newton's attitude in Principia of first con-sidering the problem in its generality, without assuming that the force obeys the gravitational inverse square law. We find that circular concentric orbits, also named relative equilibria, exist as a general fact. Conversely, we show that homographic orbits  o exist only for forces that obey a power law.

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