Acausality in Newtonian Physics

Today I read an interesting philosophy of science paper by John Norton. I have some brief (and frankly not very interesting) comments on it in my reading blog. But I wanted to share what was to me a fascinatingly surprising example that Norton gave. Norton's thesis is that philosophers of science are mistaken to understand causality as fundamental to science. He introduces this example, in his own words:

While exotic theories like quantum mechanics and general relativity violate our common expectations of causation and determinism, one routinely assumes that ordinary Newtonian mechanics will violate these expectations only in extreme circumstances if at all. That is not so. Even quite simple Newtonian systems can harbor uncaused events and ones for which the theory cannot even supply probabilities. ... Here is an example of such a system fully in accord with Newtonian mechanics. It is a mass that remains at rest in a physical environment that is completely unchanging for an arbitrary amount of time—a day, a month, an eon. Then, without any external intervention or any change in the physical environment, the mass spontaneously moves off in an arbitrary direction, with the theory supplying no probabilities for the time or direction of the motion.

If you're not excited and shocked by this point, then you're not me. Norton goes on to set up the system, in which a mass rests frictionlessly on top of a dome. He gives mathematical definitions of the dome and the force of gravity. He observes that Newton's first and second laws can be trivially solved for the mass' location at all times t, in r(t) = 0, the apex. But he also identifies a second solution class in which the mass starts moving in any radial direction after any arbitrary time T! Unfortunately, my calculus is too rusty to check the math, but I have every confidence that he's right. Again, I have nothing to say other than that I find this surprising and interesting. If you do too, check it out... he explains this system in detail in §3 of his paper, starting on page 8. There's a good diagram, too. The paper is available online as pdf or gif images of each page. If you understand the science (or the philosophy) better than I do, I'd appreciate enlightening.