1) If only positions should match, then the answer is yes. Just roll two identical balls at each other.
2) If both positions and velocities should match, then the answer is no. Here’s a sketch of a proof:
Assume without loss of generality that some balls have nonzero velocity along the X axis. Let’s define the following function of time: take all balls having nonzero X velocity, and take the lowest X coordinate of their centers. A case analysis shows that the function can change from increasing to decreasing, but never the other way. Therefore it cannot be periodic. But since it’s determined from the configuration, that means the configuration can’t be periodic, QED.
Note that the case analysis is tricky because the function can be discontinuous. The interesting cases are when a ball’s X velocity becomes zero due to a collision, or (more subtly) two balls with only Y velocity gain X velocity due to an off-center collision. But I think the statement about monotonicity still holds.
Yeah I thought about those two cases as well, but I agree that they are correct. Perhaps we could make the proof a bit simpler by picking the X direction to be one that the balls never travel perpendicular too (although in fact I can’t even think of a proof that such a direction exists).
1) If only positions should match, then the answer is yes. Just roll two identical balls at each other.
2) If both positions and velocities should match, then the answer is no. Here’s a sketch of a proof:
Assume without loss of generality that some balls have nonzero velocity along the X axis. Let’s define the following function of time: take all balls having nonzero X velocity, and take the lowest X coordinate of their centers. A case analysis shows that the function can change from increasing to decreasing, but never the other way. Therefore it cannot be periodic. But since it’s determined from the configuration, that means the configuration can’t be periodic, QED.
Yep, that looks pretty airtight to me. Well done!
Note that the case analysis is tricky because the function can be discontinuous. The interesting cases are when a ball’s X velocity becomes zero due to a collision, or (more subtly) two balls with only Y velocity gain X velocity due to an off-center collision. But I think the statement about monotonicity still holds.
Yeah I thought about those two cases as well, but I agree that they are correct. Perhaps we could make the proof a bit simpler by picking the X direction to be one that the balls never travel perpendicular too (although in fact I can’t even think of a proof that such a direction exists).
That wouldn’t help with the first case, because a ball can stop completely. Let’s keep the proof as it is :-)