# Reversibility in classical mechanics

I am watching Susskind's 'Theoretical Minimum' videos. At one point in his course on classical mechanics (2nd video if I remember correctly) he affirms that Netwon's second law of motion makes classical mechanics reversible. To make a case for it he uses the example of a spring and concludes that it is reversible.

But a spring is only one particular system amongst many and I can think of many other examples for which reversibility is not obvious. In particular, I am thinking about an object that slides on the ground with an initial velocity until friction makes it stop. When the system is in its final state, there are no way of finding which direction it came from (if it is assumed that the object leaves no traces). It could have moved from any direction of space. Therefore it is not reversible in the sense that he defined it.

What did I not understand about reversibility ? Is there a problem with the example I just gave ?

The dynamic equations of classical mechanics are locally time-reversal invariant. You can replace $t$ with $-t$ in them and they won't change their form.

A system with friction is NOT described by these equations, and that kind of system is not covered by the reversibility statement. Statistical mechanics and chaos theory give you the real arguments for the difference between local time reversal-invariance and global reversibility. For one thing you need to make the phase space finite. Here is a counterexample for what happens when you don't: imagine a frictionless ball rolling down a hill surrounded by an infinite plane. The solution is that the ball starts at zero velocity on the top of the hill and starts rolling until it reaches its final speed at the bottom. From there on it moves at constant velocity in the flat bottom of the infinite valley. We can not reverse this motion because the movement towards infinity can not be modeled with proper initial conditions. $t=-\infty, x=\infty, p=-p_0$ is not a valid set of initial conditions.

Similarly systems with an infinite degrees of freedom can not be reversed and finally systems with nonlinearities that have Lyapunov exponents make it impossible to practically reverse their trajectories after a short amount of time. As it turns out that's the case for all Hamiltonian systems but about a dozen of highly symmetric integrable systems.

What we can learn from this is that the local properties of dynamic equations of motion do not translate into global properties of their solution spaces. This opens up an amazing variety of mathematical problems which are well worth studying, even though the naive 17th-19th century notion that simple equations should lead to simple solutions is completely wrong.

• So, the conclusion is that systems involving friction are not considered classical ? Should I deduce that only systems with forces that derivates from a potential can be considered classical ? Commented Dec 27, 2014 at 13:23
• You can model friction with an ad-hoc first order velocity term in classical mechanics (or other terms that break time-reversal invariance), but those are not considered the fundamental equations of motion in classical mechanics because they don't conserve energy and there is no place in the equation where the energy goes. In essence, such an equation is "false", even though it may be a useful approximation in cases where we don't care about energy conservation. Commented Dec 27, 2014 at 14:51

Any system which can be described by newton's law. For example, if a football player kicks a ball from A to B and then he repeats from B to A (with the acceleration it had in B) the path will be the same. That means time reversal, that if t is changed by -t (and so the final position will be the initial and the inverse), the path will be the same.

It is because acceleration doesn't depends on time, only on position. As you can see $F=ma=m(\frac{dv}{dt})^2$, changing t for -t does not change Force (or acceleration), and so the path will be the same.