Theorem: If a minimizer of the fixed ends problem has an isolated $p$ body collision at $t=t_0$, then there is a parabolic homethetic collision-ejection solution which is also a minimizer of the fixed ends problem of the $p$ body subsystem .
It is considered as a powerful tool to exclude collisions. But it may be viewed from a different point. There are different types of boundary configurations, which forms different path spaces. Among those space where minimizers have collision, the path $x(t)=ct^{frac23}$ is also a minimizer of the subsystem on the subspace, where $c$ is the central configuration. From the view of critical point theory, it implies the properties of the underlying function space may reveal some important chracteristics of some minimizers.
Q: ccs may be viewed as critical points, its corresponding homographic solutions are critical points as well, then what is the relationship?