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

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.