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- W2023923021 abstract "This paper gives a proof of the existence and multiplicity of periodic solutions to Hamiltonian systems of the form [ ({text {A}})quad {text { }}left { {begin {array}{*{20}{c}} {{m_i}{{ddot q}_i} + frac {{partial V}} {{partial {q_i}}}(t,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." @default.
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- W2023923021 date "1995-12-01" @default.
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- W2023923021 title "Periodic orbits of $n$-body type problems: the fixed period case" @default.
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- W2023923021 doi "https://doi.org/10.1090/s0002-9947-1995-1316861-0" @default.
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