Abstract
The configuration space of the planar three-body problem, reduced by rotations and with collisions excluded, has a rich topology which supports a large set of free homotopy classes. These classes have a simple description in terms of syzygy (or eclipse) sequences. Each homotopy class corresponds to a unique 'reduced' syzygy sequence. We prove that each reduced syzygy sequence is realized by a periodic solution of the rotation-reduced Newtonian planar three-body problem. The realizing solutions have small, nonzero angular momentum, repeatedly come very close to triple collision, and have lots of 'stutters' - repeated syzygies of the same type, which cancel out up to homotopy. The heart of the proof stems from the work by one of us on symbolic dynamics arising out of the central configurations after the triple collision is blown up using McGehee's method. We end with a list of open problems.
Original language | English (US) |
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Pages (from-to) | 1919-1935 |
Number of pages | 17 |
Journal | Nonlinearity |
Volume | 28 |
Issue number | 6 |
DOIs | |
State | Published - Jun 1 2015 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2015 IOP Publishing Ltd & London Mathematical Society.
Keywords
- Euler solutions
- Lagrange solutions
- McGehee blow-up
- free homotopies
- planar three-body problem
- syzygy sequences
- triple collisions