Hello looking for direction, not solution. Originally I have been working with the Cuker-Smale system and we had an open space for where Momentum has been conserved in the open space, as in there have been no boundaries. I am currently trying to consider what would happen if there was a boundary or a wall on one side of a system and the alignment of momentum between collisions. Loosely speaking I am considering, show that there exists $t^*$: for all $t>t^*$ where $v_i \geq 0$ after collision hits the wall.
Materials that I am working with are:
We will approach this problem by considering the momentum $\rho$ between collisions:
Recall that $\rho$ between collisions is conserved.
At $t=0$, $\rho_0 = \frac{1}{N} \sum_{i=1}^N v_i(0)$.
$\rho(t_2^-) = \rho(t_1^+) = \frac{1}{N} \sum_{i' \in I'} |v_{i'}(t_1)| + \frac{1}{N} \sum_{i''\in I''} v_{i''}$ where $I'$ are agents that collided with the wall at $t_1$.
($\rho(t_2^-) = \rho(t_1^+)$ = agents that collide'' +agents that do not'' ).
I'm wondering how in fact I can tackle problems involving conservation of momentum with different boundaries added. Some of the agents can bounce back and forth between each other after hitting the wall, some agents can bounce off the wall and might hit the wall again before finally aligning in the one direction to model the open space case. I'm not sure when momentum reaches a maximum, it may be at $t^*$. Before there is no collision, say $\rho(t_2^-)$ = $\frac{1}{N} \sum_{i=1}^N v_i$ which is just the original momentum. But how do I consider the infinite possibilities of the agents bouncing before they align, a hint I was said was to consider $\rho$ between collisions but I find myself logically unable to consider what this means or how to show that the momentum is conserved as $t$ converges to $\infty$. Here I have written about a finite case of two such times $t$, but I still have some skepticism about how correct I am.