On the Euler collinear motion of three bodies interacting with the Newton gravitational force

Authors

  • E. Piña Universidad Autónoma Metropolitana, Unidad Iztapalapa
  • M. Alvarez-Ramírez Universidad Autónoma Metropolitana, Unidad Iztapalapa

DOI:

https://doi.org/10.31349/RevMexFis.71.020701

Keywords:

Homographic motion; central configurations; conics; few-body problem; Euler collinear; three-body problem

Abstract

We present a set of mathematical properties that are very simple notwithstanding these properties are hidden in the literature, considering the Euler case of collinear motion of three bodies, including simple mathematical properties enabling the non-specialists to become very familiar with this classical pearl of the mechanics of the three body problem.

Author Biography

M. Alvarez-Ramírez, Universidad Autónoma Metropolitana, Unidad Iztapalapa

Professor

Departamento de Matemáticas

References

L. Euler, De motu rectilineo trium corporum se mutuo attrahentium. Novi Commentarii Academiae Scientiarum Petropolitanae 11 (1767) 144-151

J.-L. Lagrange, Paris Academy Ouvres 6 (1772) 272-292, https://doi.org/10.1007/978-3-0348-0933-72

A. Wintner, The Analytical foundations of celestial mechanics (Princeton University Press, New Jersey 1947)

O. Dziobek, Über einen murkwüdingen fall des vielkörperproblems. Astron. Nach. 152 (1900) 32-46

R. Moeckel, Central configurations. In Central configurations, periodic orbits, and Hamiltonian systems, Adv. Courses Math. CRM Barcelona, (2015) 105-167, https://doi.org/10.1007/978-3-0348-0933-72

E. Piña and A. Bengochea, Hyperbolic geometry for the binary collision angles of the three-body problem in the plane Qual. Theory Dyn. Sys. 8 (2009) 399, https://doi.org/10.1007/s12346-010-0009-6

A. Albouy and R. Moeckel, The inverse problem of collinear central configurations Cel. Mech. and Dyn. Astron. 77 (2000) 77, https://doi.org/10.1023/A:1008345830461

C. Marchal, The Thee-Body Problem (Elsevier: Amsterdam 1990)

M. Álvarez Ramírez and E. P. Garza, Las configuraciones centrales en el problema restringido de 3 + 1 cuerpos en el plano. Una generalización de las ideas de Poincaré., In Hemri Poincaré y David Hilbert, Los últimos universalistas y los fundamentos de la física matemática moderna (Universidad Autónoma Metropolitana, Mexico City, 2016) pp. 127- 144

E. Piña, Three families of 5-body central configurations in the plane Cel. Mech & Dyn. Astron. 134 (2022) 43, https://doi.org/10.1007/s10569-022-10097-1

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Published

2025-03-01

How to Cite

[1]
E. Piña Garza and M. Álvarez Ramírez, “On the Euler collinear motion of three bodies interacting with the Newton gravitational force”, Rev. Mex. Fís., vol. 71, no. 2 Mar-Apr, pp. 020701 1–, Mar. 2025.

Issue

Vol. 71 No. 2 Mar-Apr (2025): Revista Mexicana de Física

Section

07 Gravitation, Mathematical Physics and Field Theory

License

Copyright (c) 2025 E. Piña, M. Alvarez-Ramírez

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