Are there solutions when the boundary conditions are particle positions at 2 different times instead of positions and speeds at an initial time?

Question

Is it possible to find solutions for a dynamic system when the boundary conditions are particle positions at 2 different times instead of positions and speeds at an initial time? The question is motivated by thoughts about the thermodynamic arrow of time. Suppose the simplest case of billiard ball particles in a gas. For instance, I do not know if I can have a solution in which half of the particles start packed together at a small place and the other half at random places, and the final positions are the first set of particles at random places and the second set packed together at some specific place. that would be a solution in which the arrows of time for the two (interacting) subsystems are in the opposite directions. It is clear that if the two subsystems do not interact you can find a solution (just take an initial system that spreads and reverse it for one of the systems, but it is not clear to me if the problem has solutions at all when the particles are allowed to interact.

Answer 1

For general dynamic systems, no you can't. Consider for instance $x^{\prime\prime} = -x$. Every solution with initial condition $x(0)=0$ will also have $x(\pi)=0$, thus boundary conditions like $x(0)=0, x(\pi)=1$ are unsolvable.

In mechanics you can use the Lagrangian formalism to try and find such solutions, but due to for instance energy conservation they still won't exist in general.

For your specific problem it suffices to note however that the (classical) laws of nature are symmetric under the operation $t\mapsto -t$. Hence any physical scenario where the balls start out packed together and end up at random places induces a physical scenario where they start out at random places and end up packed together.