# Study of the Critical Points at Infinity Arising from the Failure of the Palais-Smale Condition for n-Body Type Problems

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*Hasna Riahi*

In this work, the author examines the following: When the
Hamiltonian system \(m_i \ddot{q}_i +
(\partial V/\partial q_i) (t,q) =0\) with periodicity condition
\(q(t+T) = q(t),\;
\forall t \in \mathfrak R\) (where \(q_{i} \in \mathfrak
R^{\ell}\), \( \ell \ge 3\), \( 1 \le i \le n\),
\( q = (q_{1},...,q_{n})\) and \( V = \sum
V_{ij}(t,q_{i}-q_{j})\) with \(V_{ij}(t,\xi)\)
\(T\)-periodic in \(t\) and singular in \(\xi\)
at \(\xi = 0\)) is posed as a variational problem, the
corresponding functional does not satisfy the Palais-Smale condition
and this leads to the notion of critical points at infinity.

This volume is a study of these critical points at infinity and of the
topology of their stable and unstable manifolds. The potential
considered here satisfies the strong force hypothesis which eliminates
collision orbits. The details are given for 4-body type problems then
generalized to n-body type problems.