Global uniqueness and determinism in classical mechanics

Question

Something always bugged me about Newton's equations (or, equivalently, Euler-Lagrange/Hamilton's): Determinism, which is the philosophical framework of classical mechanics, requires that, by completely knowing the state of a system at a given instant, $\textbf{x}(t_0)$ and the law by which the system evolves, which, in dynamics, looks something like $$m\ddot{\textbf{x}}=f(\textbf{x},\dot{\textbf{x}},t)$$ You know the exact state of the system at any instant, forwards in time and, when defined, backwards. But global uniqueness theorems state that, for this to be true, the function $f$ needs some properties, namely that it doesn't "blow up" anywhere in the domain (iirc it's enough for $f$ to be uniformly continous). My question then can be posed as such: are there any systems in which the forces that naturally arise violate the global existence/uniqueness theorems? And if so, then what does this tell us about the system?

Answer 1

If a mathematical model "blows up" at some point in the future (or past) for physically-reasonable initial conditions, then we generally regard the model as being an imperfect representation of nature. The model may still be useful as an approximation for many things, but we don't expect it to be the final word, because nature shouldn't behave that way.

A famous example is the singularity theorems in general relativity. With physically-reasonable initial conditions, general relativity often predicts that a singularity will develop in the curvature of spacetime, such as the singularity that is hidden behind the event horizon of a a black hole. This is reviewed in Witten's "Light Rays, Singularities, and All That" (https://arxiv.org/abs/1901.03928). This property of general relativity is regarded as a sign that general relativity cannot be the final word: it must be just an approximation to something else, albeit an excellent approximation under less-extreme conditions. By the way, that diagnosis is consistent with a completely different type of evidence that general relativity is incomplete, namely the fact that general relativity doesn't include quantum effects. Most (all?) physicists expect that a proper quantum theory of gravity will not have such singularities — probably because the usual concept of spacetime itself is only an approximation that becomes a bad approximation in cases where non-quantum GR would have predicted a singularity.

Quantum theory isn't deterministic, but even in quantum theory, good models are required to respect a general principle called the time-slice principle, more presumptuously called unitarity. This principle says that all observables at any time in the past or future can be expressed (using sums, products, and limits) in terms of observables associated with any one time.