The answer given in Stack Exchange and the Wikipedia article linked to in one of its comments provide explanations, though whether you find them convincing is,

The answer given in Stack Exchange and the Wikipedia article linked to in one of its comments provide explanations, though whether you find them convincing is, of course, up to you! The Wikipedia article says there are solutions to the classical equations of motion: one in which the ball remains stationary forever, and then all those where, after an arbitrary period during which the ball is stationary, it rolls off the dome in an arbitrary direction. What makes this indeterminate is that the analysis of a single initial state yields multiple possible outcomes. The article goes on to say "Notice in the second case that the particle appears to begin moving without cause 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 so cannot be ruled out as non-physical." This raises the question of what we mean by 'physical', and whether theories of physics define the physical or describe something existing independently. I will leave that to the more philosophically minded; for myself, I will just note that as there are cases (and physically realizable ones at that) where classical physics gives answers that are not merely indeterminate but outright wrong (the ultraviolet catastrophe being a canonical example), I don't think anything of consequence hangs on this particular case. https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_the_equations_of_motion https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_t...