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Periodic orbits of $ n$-body type problems: the fixed period case


Author: Hasna Riahi
Journal: Trans. Amer. Math. Soc. 347 (1995), 4663-4685
MSC: Primary 58F22; Secondary 34C25, 58E05, 70F10
DOI: https://doi.org/10.1090/S0002-9947-1995-1316861-0
MathSciNet review: 1316861
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Abstract: This paper gives a proof of the existence and multiplicity of periodic solutions to Hamiltonian systems of the form

$\displaystyle ({\text{A}})\quad {\text{ }}\left\{ {\begin{array}{*{20}{c}} {{m_... ...q) = 0} \\ {q(t + T) = q(t),\quad \forall t \in \Re .} \\ \end{array} } \right.$

where $ {q_i} \in {\Re ^\ell },\ell \geqslant 3,1 \leqslant i \leqslant n,q = ({q_1}, \ldots ,{q_n})$ and with $ {V_{ij}}(t,\xi )$ $ T$-periodic in $ t$ and singular in $ \xi $ at $ \xi = 0$ Under additional hypotheses on $ V$, when (A) is posed as a variational problem, the corresponding functional, $ I$, is shown to have an unbounded sequence of critical values if the singularity of $ V$ at 0 is strong enough. The critical points of $ I$ are classical $ T$-periodic solutions of (A). Then, assuming that $ I$ has only non-degenerate critical points, up to translations, Morse type inequalities are proved and used to show that the number of critical points with a fixed Morse index $ k$ grows exponentially with $ k$, at least when $ k \equiv 0,1( \mod \ell - 2)$. The proof is based on the study of the critical points at infinity done by the author in a previous paper and generalizes the topological arguments used by A. Bahri and P. Rabinowitz in their study of the $ 3$-body problem. It uses a recent result of E. Fadell and S. Husseini on the homology of free loop spaces on configuration spaces. The detailed proof is given for the $ 4$-body problem then generalized to the $ n$-body problem.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1316861-0
Keywords: $ n$-body problems, periodic solutions, generalized Morse inequalities
Article copyright: © Copyright 1995 American Mathematical Society

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