Mathematicians Prove Symmetry and Stability in Five-Body Coorbital Configurations

Mathematicians Jiangtao Xu and Yiyang Deng have established that a special class of five-body orbital arrangements must be symmetric, in a study published in Celestial Mechanics and Dynamical Astronomy. The work concerns the planar 1+4-body problem, in which a single massive central body is orbited by four bodies whose masses tend to zero, making them effectively coorbital around the dominant mass. For the convex case, where all five bodies form a convex polygon, the researchers analyzed configurations governed by a homogeneous potential with exponent greater than one.

They proved that when the masses satisfy the symmetric conditions m1 = m4 greater than zero and m2 = m3 greater than zero, any convex coorbital central configuration must possess an axis of symmetry. Beyond the symmetry result, the authors also proved a linear stability statement: under some constraints, the symmetric convex coorbital central configuration with m1 = m4 and m2 = m3 is linearly stable. The findings apply to the mathematical model in which the four smaller masses tend to zero, so they describe a limiting regime rather than a specific planetary system.

The results are proved for the homogeneous potential setting with exponent greater than one, and the stability guarantee holds only under the stated constraints rather than for all symmetric convex configurations.

Subject of Research: Symmetry and linear stability of convex 1+4 coorbital central configurations in the planar five-body problem with homogeneous potential

Article Title: The symmetry of convex 1 + 4 coorbital problem central configurations with homogeneous potential

Article References: Xu, J., & Deng, Y. (2026). The symmetry of convex 1 + 4 coorbital problem central configurations with homogeneous potential. Celestial Mechanics and Dynamical Astronomy, 138(6), 63. https://doi.org/10.1007/s10569-026-10338-7

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DOI: 10.1007/s10569-026-10338-7

Keywords: celestial mechanics, central configurations, coorbital problem, five-body problem, relative equilibria, homogeneous potential, linear stability, symmetry, mathematical physics, orbital dynamics, convex, coorbital