Equivalence of various definitions of reversibility in classical mechanics I was reading Classical Mechanics: The Theoretical Minimum by Leonard Susskind, and the definition of reversibility in that was:

Given a state of a system, then we know exactly what state it came from, no ambiguity.

I have also heard three other definitions:

*

*If we reverse the film of a physical process, then if it's consistent with the laws of physics, then it's reversible.

*If we make the transformation $t \to -t$ then the form of physical laws remain the same.

*If we reverse the velocities of all particles, then the system retraces it steps.

I don't which of them are correct and whether they are equivalent and if some of them are equivalent then I have no idea how to prove that. Can anyone help?
 A: Your quoted definition is not about reversibility, rather about determinism.
1 and 2 are the same. 3 is a specific case to classical mechanics and Newton’s law.
To make things clearer, in a classical setting, you are typically interested in a certain number of observables, that I will regroup as one entity $O$, that depend on time. The “physical laws” are then the ordinary differential equation in time coupling your observables.
1 and 2 equivalently state that if $O(t)$ is a solution to the ODE, then so is $O(-t)$. This is reversibility. Due to the unicity of the solution given the initial conditions, this defines a transformation on the initial conditions, write it $T$. When applied to the the initial conditions of $O(t)$, it gives the initial conditions of $O(-t)$.
3 is the special case when the observables are positions $x$ and the ODE is Newton’s 2d law, namely:
$$
\ddot x=F(x)
$$
in this case, you can check that the system is reversible and that the transformation $T$ is given by fixing position and flipping the sign of velocity.
Hope this helps.
