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Central configurations of four bodies with an axis of symmetry

Abstract

A complete solution is given for a symmetric case of the problem of the planar central configurations of four bodies, when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry. The positions of the bodies on the axis of symmetry are described by angle coordinates with respect to the outside bodies. The solution is such, that giving the angle coordinates, the masses for which the given configuration is a central configuration, can be computed from simple analytical expressions of the angles. The central configurations can be described as one-parameter families, and these are discussed in detail in one convex and two concave cases. The derived formulae represent exact analytical solutions of the four-body problem.

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Acknowledgments

We thank the referees for useful suggestions.

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Correspondence to Bálint Érdi.

Appendix: The discriminant D

Appendix: The discriminant D

According to Eq. (82) and using Eqs. (78)–(80),

$$\begin{aligned} D= & {} q^2-4pr =\big [a_1(2b_0-b_1)+(b_0-a_0)(2a_0-a_1)\big ]^2(a_1+b_0-a_0)^2 \\&+\,4a_0(a_1+b_0-a_0)^2(2a_0-a_1+b_1-2b_0)(a_0b_1+a_1b_0-a_1b_1)\\= & {} (a_1+b_0-a_0)^2D_1, \end{aligned}$$

where

$$\begin{aligned} D_1= & {} \big [a_1(2b_0-b_1)+(b_0-a_0)(2a_0-a_1)\big ]^2\\&+\,4a_0(2a_0-a_1+b_1-2b_0)(a_0b_1+a_1b_0-a_1b_1) \\= & {} \big [a_1(2b_0-b_1)+b_1(2a_0-a_1)+(b_0-a_0-b_1)(2a_0-a_1)\big ]^2 \\&+\,4a_0(2a_0-a_1+b_1-2b_0)(a_0b_1+a_1b_0-a_1b_1) \\= & {} \big [2a_1b_0-a_1b_1+2a_0b_1-a_1b_1+(b_0-a_0-b_1)(2a_0-a_1)\big ]^2 \\&+\,4a_0(2a_0-a_1+b_1-2b_0)(a_0b_1+a_1b_0-a_1b_1) \\= & {} \big [2(a_0b_1+a_1b_0-a_1b_1)-(b_1+a_0-b_0)(2a_0-a_1)\big ]^2 \\&+\,4a_0(2a_0-a_1+b_1-2b_0)(a_0b_1+a_1b_0-a_1b_1) \\= & {} 4(a_0b_1+a_1b_0-a_1b_1)^2+(b_1+a_0-b_0)^2(2a_0-a_1)^2 \\&-\,4(a_0b_1+a_1b_0-a_1b_1)(b_1+a_0-b_0)(2a_0-a_1) \\&+\,4a_0(2a_0-a_1+b_1-2b_0)(a_0b_1+a_1b_0-a_1b_1) \\= & {} 4(a_0b_1+a_1b_0-a_1b_1)^2+(b_1+a_0-b_0)^2(2a_0-a_1)^2 \\&+\,4(a_0b_1+a_1b_0-a_1b_1)\big [-(b_1-b_0)(2a_0-a_1) +a_0(b_1-2b_0)\big ] \\= & {} 4(a_0b_1+a_1b_0-a_1b_1)\big [a_0b_1+a_1b_0-a_1b_1-(b_1-b_0)(2a_0-a_1) \!+\!a_0(b_1-2b_0)\big ] \\&+\,(b_1+a_0-b_0)^2(2a_0-a_1)^2 \\= & {} 4(a_0b_1+a_1b_0-a_1b_1)(a_0b_1+a_1b_0-a_1b_1-2a_0b_1+2a_0b_0+a_1b_1-a_1b_0\\&+\,a_0b_1-2a_0b_0) \\&+\,(b_1+a_0-b_0)^2(2a_0-a_1)^2=(b_1+a_0-b_0)^2(2a_0-a_1)^2. \end{aligned}$$

Thus

$$\begin{aligned} D=\big [(2a_0-a_1)(a_1+b_0-a_0)(b_1+a_0-b_0)\big ]^2. \end{aligned}$$

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Érdi, B., Czirják, Z. Central configurations of four bodies with an axis of symmetry. Celest Mech Dyn Astr 125, 33–70 (2016). https://doi.org/10.1007/s10569-016-9672-5

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Keywords

  • Four-body problem
  • Central configurations
  • Analytical solutions