The problem of infinite Spin for parabolic and collision solutions in the planar $n$-body problem

In the planar $n$-body problem, the problem of infinite spin occurs for both parabolic and collision solutions. Recently Moeckel and Montgomery \cite{MM25} showed that there is no infinite spin for total collision solutions, when the reduced and normalized configuration converges to an isolated central configuration. Following their approach, we show it can not happen for both complete and partially parabolic solutions, under similar conditions. Our approach also allows us to generalize Moeckel and Montgomery’s result to partial collision solutions under similar conditions.