| Abstract: |
| The figure-eight choreography and the equilateral triangle solution are two remarkable periodic motions in the equal-mass three-body problem. In 2000, C.~Marchal conjectured that they belong to the same continuation class, namely the highly symmetric $P_{12}$ family of relative periodic solutions in a rotating frame. In this talk I will describe a computer-assisted proof of this conjecture.
After exploiting the symmetries of the problem, the search for choreographic solutions reduces to a delay differential equation for a single body in rotating coordinates. The formulation is singular at both ends of the family, corresponding to the figure-eight and the Lagrange triangle, so the proof combines symmetry reduction with suitable desingularizations at the two endpoints. The main step is an a posteriori validation argument proving the existence of an analytic branch of periodic solutions on the whole parameter interval. The proof uses validated numerics, Fourier-Chebyshev expansions, and interval arithmetic.
This is joint work with Olivier H\`enot, Carlos Garc\`ia-Azpeitia, Jean-Philippe Lessard, and Jason Mireles James. |
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