Post-Newtonian effects on Lagranges equilateral triangular solution for the three-body problem

Continuing work initiated in earlier publications [Yamada, Asada, Phys. Rev. D 82, 104019 (2010), 83, 024040 (2011)], we investigate the post-Newtonian effects on Lagrange’s equilateral triangular solution for the three-body problem. For three finite masses, it is found that the equilateral triangular configuration satisfies the post-Newtonian equation of motion in general relativity, if and only if all three masses are equal. When a test mass is included, the equilateral configuration is possible for two cases: (1) one mass is finite and the other two are zero, or (2) two of the masses are finite and equal, and the third one is zero, namely a symmetric binary with a test mass. The angular velocity of the post-Newtonian equilateral triangular configuration is always smaller than the Newtonian one, provided that the masses and the side length are the same.

💡 Research Summary

This paper revisits the classical Lagrange equilateral‑triangle solution of the three‑body problem within the framework of general relativity by applying the first post‑Newtonian (1PN) approximation. Building on earlier work by Yamada and Asada (Phys. Rev. D 82, 104019 (2010); 83, 024040 (2011)), the authors employ the Einstein‑Infeld‑Hoffmann (EIH) equations of motion, which expand the gravitational interaction to order c⁻², to examine whether the equilateral configuration can still satisfy the relativistic equations of motion.

The analysis proceeds by inserting the geometric constraints of an equilateral triangle (all side lengths equal to a) into the 1PN acceleration expressions for three point masses m₁, m₂, and m₃. The resulting conditions separate into two distinct categories. First, the fully symmetric case m₁ = m₂ = m₃ is shown to be the only configuration with three finite masses that preserves the equilateral shape at 1PN order. In this situation every PN correction term appears symmetrically for each body, and the net force balance reduces to a simple generalisation of the Newtonian condition. Second, when one or more masses are taken to be infinitesimally small (test masses), the PN terms associated with those bodies vanish. Consequently two additional possibilities arise: (i) a single finite mass with the other two masses set to zero, and (ii) a symmetric binary (two equal finite masses) together with a test mass. In both cases the remaining finite masses form an equilateral triangle with the test particle, and the 1PN equations of motion are satisfied.

A key quantitative result concerns the orbital angular velocity. For a Newtonian equilateral triangle the angular speed is ω_N = √(GM/a³), where M = m₁ + m₂ + m₃. The authors derive the 1PN correction to this frequency as
 ω = ω_N