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Infinitely many coplanarities

Hamiltonian systems, from topology to applications through analysis II November 26, 2018 - November 30, 2018

November 26, 2018 (09:30 AM PST - 10:30 AM PST)
Speaker(s): Richard Montgomery (University of California, Santa Cruz)
Location: MSRI: Simons Auditorium
  • Shape space. Principal Value Decomposition

  • Newton's equations

  • N-body problem

  • Hamilton-Jacobi equations

  • Sign and the Meaning of Curvature

Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification



Consider the Newtonian 4-body problem in 3-space with zero angular momentum and any mass ratios. Theorem: every bounded solution defined on an infinite time interval suffers infinitely many coplanar instants: instants where all four bodies lie in the same plane. This generalizes a theorem asserting that for the zero angular momentum 3-body problem in the plane, every bounded solution suffers infinitely many collinear instants, or ``syzygies''. The obvious generalization holds for any dimension $d$, that is, for the zero angular momentum $d+1$-body problem in $d$-space. After stating the results, I give a heuristic physical ``proof'' of this theorem inspired by Mark Levi. I then proceed to sketch the mathematical proof. Among its main ingredients are that (i) the translation-reduced problem has for its configuration space the space of $d \times d$ matrices (ii) the signed distance from the degeneration locus satisfies a Hamilton-Jacobi equation, (iii) the oriented shape space is homeomorphic to a Euclidean space (iv) the oriented shape space has non-negative sectional curvature where smooth. The theorem gives a ``hunting license'' to set up symbolic dynamics for the spatial 4-body problem, in analogy with the idea of syzygy sequences for the planar three-body problem.

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