Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit

DOI10.1007/s00220-009-0860-yzbMath1231.37031OpenAlexW2008729046MaRDI QIDQ1048065

Publication date: 11 January 2010

Published in: Communications in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00220-009-0860-y

zbMATH Keywords

$n$-body problem Hamiltonian system Maslov index symmetric periodic orbits

Mathematics Subject Classification ID

Lagrangian submanifolds; Maslov index (53D12) (n)-body problems (70F10) Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics (70H12)

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Cites Work