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.