Randomness v. complexity

Question

There are a few other topics I found that explore this idea from a different perspective:

• Is randomness deterministic?
• Can randomness exist?
• Is the universe fundamentally deterministic?

My question is on similar lines but from the perspective of complexity and within the framework of classical mechanics. It would seem to me that what we typically describe as "random", for example in probability theory, refers to the unpredictability within a complex system. A coin flip has many different initial conditions: the size and shape of the coin, the way the hand moves when it flips it, the material of the surface the coin lands on, the movement of molecules in the air etc. It also has many many interacting parts, many of which exist on the atomic scale, and whose relationship with other parts change dynamically. Probability is a mathematical model that allows us to tell what possible average outputs the system can produce given all the internal variables that are unknown to us. Is this a fair characterization?

However, if we were, in theory, to know all the initial conditions and had a way of describing all of the interactions of all variables in this dynamic system once it has begun, could we not predict the outcome (assuming theoretical computational power)?

My main question: Is there any kind of dynamic system in classical mechanics in which the outcome would not be predictable and, if so, why not?

Answer 1

Other answers have pointed to the fact that the question in classical mechanics has only limited applicability in real life (CuriousOne's comment in particular points out that this is at best an academic question), but let's discuss it out of curiosity nevertheless.

Most systems in classical mechanics are deterministic. Here is a simple heuristic: The equations of motion are governed by Newton's laws. If the forces do not depend on the time derivatives, this implies that the equations of motions can be seen as ordinary differential equations of second order. Those can be made into a system of first order differential equations with twice as many variables (the other set of variables is the velocities). Now the theorem of Picard-Lindelöff tells us that given initial conditions, i.e. positions and velocities, the trajectory is usually (meaning when the terms of the theorem are met) unique. This uniqueness is exactly what is meant by "determinism".

For most systems, the conditions of Picard-Lindelöff will hold almost everywhere because of general assumptions about the continuity of forces, however it is possible to construct examples where it fails and where explicitly non-unique solutions can be constructed. One such example is known as Norton's dome.

Qmechanic's beautiful answer on another StackExchange question illuminates why the conditions of Picard-Lindelöff are not met and why this system is indeed not deterministic.

The idea is the following: Given the potential $h(r)=−\frac{2}{3g}r^{3/2}$ and gravity, at the top of this dome, there are at least two solutions of the equation as demonstrated in this answer.

However, there is even another class of solutions (and I'll paraphrase Norton himself from now on):

$$h(t)=\begin{cases}(1/144) (t-T)^4 \quad t>T \\ 0 \quad t \leq T \end{cases}$$

In this solution, the ball at the top of the dome suddenly starts moving and this "suddenly" $T$ can be chosen as you wish. This of course violates determinism.

Why does it work? One heuristic is also given: Newton's equations are time reversible and the solution is the time reversed version where you start from the bottom of the dome and then move upwards and the particle has just enough momentum to reach the top, but not more. Since you can find many trajectories through the top of the dome with zero velocity at the top, this would indicate that we might have a uniqueness problem there.

But beware: It doesn't prove anything, because the same is true if you just consider the top of a hill shaped like a ball half - yet here we don't have a problem with determinism, as one can prove from the initial value problem. The reason is that all trajectories through the top of the hill with zero velocity at the top only reach the top at infinite times. This is NOT true for the Norton dome. Here, the ball reaches the top at finite times!