| Abstract: |
| We present a systematic approach for proving the existence of spatial
choreographies in the gravitational $n$ body problem. After changing to
rotating coordinates and exploiting symmetries, the equation of a
choreographic configuration is reduced to a delay differential equation
(DDE) describing a single body. We study
periodic solutions of this DDE in a Banach space of rapidly decaying Fourier
coefficients. Our
argument is constructive and makes extensive use of the digital computer. Also, we
present a recent progress in the proof of a conjecture by Marchal regarding a
family of periodic solutions connecting the Lagrangre triangle to the figure
eight choreography. |
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