How do you know the classical deterministic past when particles bind?

Question

Firstly, this is assuming a classical mechanics approach as outlined in Leonard Susskind's theoretical minimum (first lecture).

In a classical setting assuming no quantum effects, how do you tell the past when two particles bind. Susskind posits that for a physical law to be viable in a classical sense is that if you know all states and what they are doing in a closed system you can tell what will happen, and also exactly what was happening before (ad infinitum).

However, this makes sense with particles repelling (in sense of elastic rebounding), but what about when particles attract? If you have a particle from 5m away versus 10m away, how do you know how far it was if velocity is lost upon binding?

Answer 1

You know how far away they were because energy is conserved. Even if they stick together completely after attracting, the internal energy of these particles must be increased. You can measure the internal energy of these particles through, say, measuring their temperature (via their blackbody spectrum or something similar). Therefore, you can determine how much energy was involved in the collision.

You can also determine the impact parameter of the collision (and also the plane in which the collision occurred) by measuring the angular momentum of the final system. The velocity of the center of mass gives you the sum of the initial momenta. The phase of the rotation of the system then gives you the particles' initial direction of approach, which means you can completely reconstruct the initial conditions.