# Oscillating about coplanarity in the 4 body problem

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## Abstract

For the Newtonian 4-body problem in space we prove that any zero angular momentum bounded solution suffers infinitely many coplanar instants, that is, times at which all 4 bodies lie in the same plane. This result generalizes a known result for collinear instants (“syzygies”) in the zero angular momentum planar 3-body problem, and extends to the \(d+1\) body problem in *d*-space. The proof begins by identifying the translation-reduced configuration space with real \(d \times d\) matrices, the degeneration locus (set of coplanar configurations when \(d=3\)) with the set of matrices having determinant zero, and the mass metric with the Frobenius (standard Euclidean) norm. Let *S* denote the signed distance from a matrix to the hypersurface of matrices with determinant zero. The proof hinges on establishing a harmonic oscillator type ODE for *S* along solutions. Bounds on inter-body distances then yield an explicit lower bound \(\omega \) for the frequency of this oscillator, guaranteeing a degeneration within every time interval of length \(\pi /\omega \). The non-negativity of the curvature of oriented shape space (the quotient of the translation-reduced configuration space by the rotation group) plays a crucial role in the proof.

## Notes

### Acknowledgements

I would like to thank Alain Albouy, Gil Bor, Joseph Gerver, Connor Jackman, Adrian Mauricio Escobar Ruiz, Robert Littlejohn, Rick Moeckel and Albert Fathi for useful discussions. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester.

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