What situations in classical physics are non-deterministic? (2018)

Chaos vs. Determinism

• Several commenters stress that classical chaos (three‑body problem, coupled/compound pendulums, molecular collisions) is still deterministic: identical initial conditions produce identical trajectories.
• Chaos limits predictability due to exponential sensitivity to initial conditions and finite measurement precision, but does not introduce fundamental non‑uniqueness.
• Some confusion appears where “stochastic” or “chaotic” is conflated with “non‑deterministic”; others explicitly correct this.

Norton’s Dome: Claimed Classical Non‑Determinism

• The dome is shaped so a particle can be rolled up and come to exact rest at the apex in finite time.
• By time‑reversal symmetry of Newtonian mechanics, there exist mathematically valid trajectories where a particle sits motionless at the apex for an arbitrary time, then spontaneously rolls off in any direction.
• Crucial point: multiple distinct trajectories satisfy the same initial state (position and velocity zero at the top) and the same Newtonian equation of motion ⇒ non‑unique evolution under the model.

Mathematical Subtleties and Critiques

• The differential equation for the dome violates the usual Lipschitz condition required for the standard existence‑and‑uniqueness theorem for ODEs; this is why multiple solutions are allowed.
• Some argue the “paradox” is just exploiting incomplete axiomatization of classical mechanics: if one assumes forces always give unique second‑order ODE solutions, Norton’s dome is simply disallowed by definition.
• Others highlight related math facts: you can have smooth functions with all derivatives zero at a point yet non‑zero elsewhere, undercutting naive “all derivatives zero ⇒ never moves” intuition.
• Skeptical voices call the construction “nonsense,” “sleight of hand,” or analogous to division by zero: a breakdown of the model rather than proof of physical indeterminism.

Physical Realizability and Idealization

• Debate over whether the dome “exists in nature”: perfect shapes, infinitesimal contact points, infinite curvature at the apex, and absence of thermal motion or air currents are all unrealistic.
• One camp: as with perfect spheres or cubes, idealized shapes are legitimate objects in the classical model; the question is about the model’s determinism, not real materials.
• Opposing camp: if constructing such a shape is impossible and the pathology is destroyed by any small perturbation, it should not count as a serious physical counterexample.

Thermodynamics, Stochastic Models, and Emergence

• Heat transfer and thermodynamics often use stochastic descriptions, but several comments emphasize this can emerge from underlying deterministic particle dynamics.
• Macroscopic randomness and entropy are framed as emergent/statistical, not necessarily fundamental non‑determinism in classical mechanics.

Quantum vs. Classical Non‑Determinism; Model vs. Reality

• Commenters distinguish classical “multi‑solution” non‑determinism (model under‑specification) from genuinely probabilistic evolution posited by some quantum interpretations.
• Some suggest that if a classical model admits non‑deterministic solutions (Norton’s dome, “space invaders”), that might instead signal those configurations are unphysical or the model incomplete.