Norton's dome is a thought experiment that exhibits a non-deterministic system within the bounds of Newtonian mechanics. It was devised by John D. Norton and first discussed in his 2003 paper "Causation as Folk Science". Norton's dome problem can be regarded as a problem in physics, mathematics, or philosophy. It poses interesting philosophical questions about the concepts of causality, determinism, and probability theory.
The model consists of an idealized particle initially sitting motionless at the apex of an idealized radially symmetrical frictionless dome described by the equation
where h is the vertical displacement from the top of the dome to a point on the dome, r is the geodesic distance from the dome's apex to that point (in other words, a radial coordinate r is "inscribed" on the surface), and g the acceleration due to gravity.
Norton shows that there are two classes of mathematical solutions to the system under Newtonian physics. In the first, the particle stays sitting at the apex of the dome forever. In the second, the particle sits at the apex of the dome for a while, and then after an arbitrary period of time starts to slide down the dome in an arbitrary direction. The apparent paradox in this second case is that this would seem to occur for no discernible reason, and without any radial force being exerted on it by any other entity, apparently contrary to both physical intuition and normal intuitive concepts of cause and effect, yet the motion is still entirely consistent with the mathematics of Newton's laws of motion.
While many criticisms have been made of Norton's thought experiment, such as it being a violation of the principle of Lipschitz continuity, or in violation of the principles of physical symmetry, or that it is somehow in some other way "unphysical", there is no consensus among its critics as to why they regard it as invalid.