In the N-body problem, choreographies are periodic solutions where N equal masses follow each other along a closed curve. Each mass takes periodically the position of the next after a fixed interval of time. In 1993, Moore discovered numerically a choreography for N = 3 in the shape of an eight. The proof of its existence is established in 2000 by Chenciner and Montgomery. In the same year, Marchal published his work on the most symmetric family of spatial periodic orbits, bifurcating from the Lagrange triangle by continuation with respect to the period. This continuation class is referred to as the P12 family. Noting that the figure eight possesses the same twelve symmetries as the P12 family, the author claimed that it ought to belong to P12. This is known as Marchal’s conjecture. In this talk, we present a constructive proof of Marchal’s conjecture. We formulate a one parameter family of functional equations, whose zeros correspond to periodic solutions satisfying the symmetries of P12; the frequency of a rotating frame is used as the continuation parameter. The goal is then to prove the uniform contraction of a mapping, in a neighbourhood of an approximation of the family of choreographies starting at the Lagrange triangle and ending at the figure eight. The contraction is set in the Banach space of rapidly decaying Fourier-Chebyshev series coefficients. While the Fourier basis is employed to model the temporal periodicity of the solutions, the Chebyshev basis captures their parameter dependence. In this framework, we obtain a high-order approximation of the family as a finite number of Fourier polynomials, where each coefficient is itself given by a finite number of Chebyshev polynomials. The contraction argument hinges on the local isolation of each individual choreography in the family. However, symmetry breaking bifurcations occur at the Lagrange triangle and the figure eight. At the figure eight, there is a translation invariance in the normal direction to the eight. We explore how the conservation of the linear momentum in this direction can be leveraged to impose a zero average value in time for the choreographies. Lastly, at the Lagrange triangle, its (planar) homothetic family meets the (off-plane) P12 family. We discuss how a blow-up (as in “zoom-in”) method provides an auxiliary problem which only retains the desired P12 family.
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