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In Sean Carroll's book "The Big Picture," he states (chapter 4, page 35):

Classical mechanics, the system of equations studied by Newton and Laplace, isn't perfectly deterministic. There are examples of cases where a unique outcome cannot be predicted from the current state of the system. This doesn't bother most people, since cases like this are extremely rare—they are essentially infinitely unlikely among the set of all possible things a system could be doing. They are artificial and fun to think about, but not of great import to what happens in the messy world around us.

What are some of these situations he is referring to?

My first guess was something like the conceptual equivalent of a ball sitting exactly atop a hill and asking which direction it will roll. But that doesn't seem right—this would be a case where the "true" solution is that the ball wouldn't roll at all, while in reality there will always be tiny disturbances that cause the ball to roll in some direction. I don't think this is what Carroll is referring to. He seems to suggest that the "true, exact mathematical outcome" is not uniquely determined for certain classical systems.

I've heard that uniqueness theorems for solutions to differential equations don't generally apply to non-linear equations, and wonder if this may have something to do with it. Elucidation and examples would be much appreciated.

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    $\begingroup$ en.wikipedia.org/wiki/Norton%27s_dome $\endgroup$ – AccidentalFourierTransform May 2 '18 at 21:14
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    $\begingroup$ The evolution of chaotic systems cannot be predicted from their initial conditions. $\endgroup$ – rob May 2 '18 at 21:17
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    $\begingroup$ Related: Norton's dome and its equation. $\endgroup$ – Qmechanic May 2 '18 at 21:23
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    $\begingroup$ @rob, note that most cases of chaos in classical mechanics are deterministic chaos. Only specific examples would be non-deterministic but chaos itself does not mean that the system is non-deterministic. $\endgroup$ – fhorrobin May 2 '18 at 21:47
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    $\begingroup$ Models based on classical mechanics are not necessarily computable. An explicit example was given a long time ago involving a system that implements a computer that accelerates without limit such that each clock cycle takes half the time as the previous one. This means that an infinite number of computations can be performed in a finite amount of time. Such a system can then solve non-computable mathematical problems, but that then means that the state of the system is itself non-computable. $\endgroup$ – Count Iblis May 2 '18 at 22:00
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There are two famous cases in classical mechanics that fail to be deterministic.

The first, and most famous, is Norton's Dome, which corresponds to a system with a force of the form

$$F = \sqrt{r} $$

There are more details on the Wikipedia article (it's usually described as the result of a reaction force from a surface with a certain shape), but the basic idea is that the derivative of the force fails to be defined at $r = 0$, since

$$(\sqrt{r})' = \frac{1}{2\sqrt{r}}$$

Due to this, there's no guarantee that the equation $\ddot{r} = \sqrt{r}$ has a unique solution (and indeed it doesn't), because it fails to be Lipschitz continuous.

There's plenty of informations about Norton's Dome, both here and on the internet, so here's the more interesting, if even more pathological, example, the Space Invader.

The space invader is a particle which is submitted to an unbounded acceleration in finite time, so that it reaches "infinity" after a while. The exact form of the force doesn't matter, but for instance you could pick

$$F = \tan(t)$$

In such cases, the particle will go off to infinity at $t = \pi/2$ and, after that time, cease to exist. As this system is time-symmetric, it is also possible to consider the case of a particle which originally does not exist and comes from infinity, or even doing both (restriction of the force to specific time intervals will do to insure those outcomes).

Another example of such a behaviour are the Painlevé non-collision singularities. The most famous example of which is a 5-body gravitational problem where one of the particle will also go to infinity in finite time, by simply borrowing energy from two 2-body systems. As for point-particles, the potential energy is unbounded from below (since it is $E \propto -1/r$), it is possible to have an infinite kinetic energy while maintaining conservation of energy, by having the 2-body systems in it collapse.

For a general treatment on the issue of determinism in classical physics, you can also check this article of Earman, for instance.

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  • $\begingroup$ how would Norton's dome behave if we introduced an arbitrarily small but nonzero friction force? $\endgroup$ – hyportnex May 2 '18 at 22:59
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    $\begingroup$ @hyportnex you would still lack Lipschitz continuity, so the conclusion would most likely be the same. $\endgroup$ – AccidentalFourierTransform May 3 '18 at 1:55
  • $\begingroup$ Did you mean Painlevé non- collision singularities? $\endgroup$ – Ruslan May 3 '18 at 7:17
  • $\begingroup$ that is indeed the correct one. $\endgroup$ – Slereah May 3 '18 at 8:21

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