3 ms·

We don't have a term for this, but it's like an inertia for forces acting on the particle (gravity and the surface normal). You can also view this as a missing

We don't have a term for this, but it's like an inertia for forces acting on the particle (gravity and the surface normal). You can also view this as a missing piece in Newton's laws, as this is not explicitly covered as a possibility. But that is also true then for the entire motion, as all Newtonian equations of motion have the derivatives of acceleration equal to 0, while the motion along this surface necessarily doesn't.

wat10000 2y ago
No such term exists in Newton’s laws. If a particle accelerates, it must be because of a force acting on it. Not something “like an inertia for forces,” but an actual, physical force. Two forces act on this particle, but they sum to zero. If you claim the particle begins to move from zero velocity, then either those two forces don’t sum to zero, or there’s a third force. But they do and there isn’t.
- tsimionescu 2y ago
  Yes, this "inertia-of-force" is not part of Newton's laws, but it's a straightforward extension of Newton's laws, not a contradiction. And it preserves all of the other interesting properties - energy conservation, time symmetry, determinism. It's certainly less of a contradiction of Netwon's laws than the idea that the particle just sitting there can suddenly start moving, which is a straightforward contradiction of Netwon's first law, and implies that a particle can acquire higher order acceleration (jerk or snap) out of thin air. Of course, this is all moot anyway, as this type of dome with a point-like apex is not a physical system to begin with: since matter is made of atoms, it can't have this type of singular point. Not even going into problems of the precision needed to give the particle just the right initial speed to actually have exactly 0 velocity at the top.
  - wat10000 2y ago
    Inventing a different physics and declaring it to be more correct certainly works, but is kind of a boring answer to the problem. It plainly is a contradiction given that there’s a system where it gives a different answer. If it wasn’t a contradiction then it would be equivalent (or somehow a superset?) and wouldn’t solve the problem. Ignoring the problem as unphysical makes much more sense to me. The real world doesn’t follow Newtonian physics anyway, it’s effectively just a simplification useful for certain common situations. No need to care about edge cases that can’t happen.

wat10000 2y ago

No such term exists in Newton’s laws. If a particle accelerates, it must be because of a force acting on it. Not something “like an inertia for forces,” but an actual, physical force. Two forces act on this particle, but they sum to zero. If you claim the particle begins to move from zero velocity, then either those two forces don’t sum to zero, or there’s a third force. But they do and there isn’t.

tsimionescu 2y ago
Yes, this "inertia-of-force" is not part of Newton's laws, but it's a straightforward extension of Newton's laws, not a contradiction. And it preserves all of the other interesting properties - energy conservation, time symmetry, determinism. It's certainly less of a contradiction of Netwon's laws than the idea that the particle just sitting there can suddenly start moving, which is a straightforward contradiction of Netwon's first law, and implies that a particle can acquire higher order acceleration (jerk or snap) out of thin air. Of course, this is all moot anyway, as this type of dome with a point-like apex is not a physical system to begin with: since matter is made of atoms, it can't have this type of singular point. Not even going into problems of the precision needed to give the particle just the right initial speed to actually have exactly 0 velocity at the top.
- wat10000 2y ago
  Inventing a different physics and declaring it to be more correct certainly works, but is kind of a boring answer to the problem. It plainly is a contradiction given that there’s a system where it gives a different answer. If it wasn’t a contradiction then it would be equivalent (or somehow a superset?) and wouldn’t solve the problem. Ignoring the problem as unphysical makes much more sense to me. The real world doesn’t follow Newtonian physics anyway, it’s effectively just a simplification useful for certain common situations. No need to care about edge cases that can’t happen.