对venturelli theorem的重新认识

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?