Scientists find an effective solution for the three-body problem

> There is no confusion here. A singularity exists at the top of Nortons Dome and Norton is exactly describing the behavior of the particle AFTER it exits the singularity. The exact same phenomena is occurring in the n-body problem as nortons dome.

There's no singularity in Norton's dome. The equation d^2r/dt^2 = r^{1/2} is well-defined for all non-negative times t. This is not the case in the three-body problem, where a coefficient blows up in finite time when two bodies collide. The dome example is not relevant here and you are confusing yourself by conflating the two phenomena.

>> >It doesn't make sense to talk about a solution past the point of collision. One doesn't exist.

> This is not true.

And just to repeat myself: it is true, it's a mathematical theorem, you can't continue the solution past the point of the collision. Please check the references.

If you want a heuristic reason for this, note that conservation of energy implies infinite velocities at the point of collision. The situation is inherently unphysical due to it being a mathematical idealization.

edit: If you're serious about the $1k thing, I propose the following. We agree on a precise mathematical formulation of the question "Is the three-body problem deterministic?," an expert, and a charity. We email the expert for comment. If I am right, you donate to the charity. If I am wrong, I will admit to it here, and perhaps donate some money myself. We can put the agreement on our personal websites and post it to HN to publicly commit.

And I have countered your argument. I get it. You're trying to say that we're talking past each other here and that you're just repeating yourself. No. I addressed your point, you don't need to repeat yourself you only need to counter the new point.

That being said, there is no new error. See this paper here which talks about the singularities at the top of the dome:

http://philsci-archive.pitt.edu/3195/1/NortonDome.pdf

Norton is 100% dealing with a singularity and therefore he is describing the exact same phenomenon of what happens to the particle after it exits the singularity.

>And just to repeat myself: it is true, it's a mathematical theorem, you can't continue the solution past the point of the collision. Please check the references.

I never said the mathematical theorem is not true. I said it doesn't apply. I did check the references. The theorem only applies for certain conditions and those conditions aren't met, so the theorem has nothing to say about the system at that point.

Let's be clear. The theorem says one thing and one thing only that a unique solution exists if the conditions are met. When the conditions are not met (at the singularity) the theorem says absolutely nothing and therefore it does not apply. The theorem does not say that the solution does not exist, it just doesn't apply.

>If you want a heuristic reason for this, note that conservation of energy implies infinite velocities at the point of collision. The situation is inherently unphysical due to it being a mathematical idealization.

Doesn't matter. Infinite velocities is something you can't consider because the model is undefined at the singularity but it is defined both before and after the singularity. To give you a heursitic reason as well: imagine the successor function for peano arithmetic in number theory.

We have a number X where X = 0 and subsequent numbers are defined as S(X) or S(S(X))) and so on.

If X = 0 exists and A is any number, then A/S(X) still exists even though A/X doesn't exist (division by zero). This is what's going on here. Energy is conserved at the singularity (X = 0) but what that implies is undefined (1/X). But outside of the singularity the implication 1/S(X) is defined. There's no rule saying that 1/S(X) can't be defined just because 1/X is not defined. We are still in the purely mathematical realm here, we are not talking about physics in reality.

Where the mechanics becomes nondeterministic is that (to continue with the heuristic analogy from above) 1/S(X) where X=0 is nondeterministic. Just like in number theory The literal math states that the system is undefined at the singularity. In other words the math DEFINES the solution to be UNDEFINED at the singularity. However the math itself makes zero statement about the system AFTER the particle exits the singularity. Hence we can say certain things about the particle if/when it exits the singularity even though many things are unknown. This is the reason why Norton is able to arrive at his conclusion of bounded nondeterminism. Certain things can be derived but newtons laws about the particle at 1/S(X) but not enough to arrive at a unique nondeterministic solution. The exact same thing applies to the n-body problem as it does to Nortons Dome.

Also take a look at the paper I linked above and read the "Space Invaders" section. It addresses another possibility of in-determinism in a Newtonian n-body system based on how we arbitrarily choose to define it. This topic alone will initiate another angle in which to approach the question of determinism of an N-body system.

>edit: If you're serious about the $1k thing, I propose the following. We agree on a precise mathematical formulation of the question "Is the three-body problem deterministic?," an expert, and a charity. We email the expert for comment. If I am right, you donate to the charity. If I am wrong, I will admit to it here, and perhaps donate some money myself. We can put the agreement on our personal websites and post it to HN to publicly commit.

I am serious. The problem with experts on this topic is they all have different opinions on the topic. What if I pick the expert and author of the paper I linked above? There is no consensus among experts. I'll think about it but I don't want to be in a situation where the expert picks a solution and I remain unconvinced only to later find another expert who has the opposite opinion.

I will acknowledge that there's an obvious $1000 bias on my side here and that a 3rd party levels the playing ground but unfortunately there is heavy bias on the "expert" side as well. I mean aren't you an "expert"? Hence your confidence on this topic.

I think for now, unfortunately, we're going to have to settle with you convincing me and you'll have to take my word that if you can convince me I'll concede and try to look past my biases. Either way I'll think about your proposal.

Also you should note my initial bet post was flagged so this entire thread is basically dead except for a couple people still responding to me.

https://news.ycombinator.com/newsguidelines.html

https://en.wikipedia.org/wiki/Functor. The functor is a generalization of the concept of changing "space". It specifically refers to the mapping between these spaces. For example changing from json to xml, or changing from cartesian coordinates to polar coordinates, euler angles to quaternions or changing from Newtonian mechanics to Lagrangian.

I am saying there's no point in using a functor if the point is already proven. You gain no ground and telling me to change space because of entropy. Things may be easier in the secondary space but because of information entropy there may or may not be a loss of information and this loss definitively leads to less capability of proving anything.

>I did, and he does not. The fact is that in Newtonian mechanics, an object at rest cannot start moving without a change in the applied forces. By definition, if it does, then it does not follow Newtonian mechanics.

Doubtful as you didn't even address his point. You stated your point as if his counterpoint didn't even exist. Either way a discontinuity is undefined but it is 100% legal to talk about the points where it is defined. The limits are legal to address mathematically as there is no math stating that those points are singularities or non-existent or anything like that. You statement of it being unfounded doesn't move your argument in any direction. You need to prove your point or disprove Nortons point.

>But that would not happen, because it is non-Newtonian. What he does in fact demonstrate is that a ball with exactly the right energy does not arrive at the apex in a finite time. He got the contradiction right, but sided the wrong way. Besides, he even mentions himself that the ball would not arrive in a finite time, and we are supposed to believe that this trajectory is the time-inversion image of a ball that definitely leaves the apex in a finite time.

Maybe instead of reading the entire article really quickly and missing the entire point you should read it more carefully. He is talking about the spherical dome. The ball not arriving at finite time is for the perfect hermsphere. For Nortons Dome such an action is 100% possible under newtons model.

>>The whole dome setup is a troll. There is nothing in the principles he mentions that would not work with an ordinary, half-spherical dome, if it did in fact work. His specific dome sounds suspiciously like an artificial setup to get people hung up in irrelevant mathematical details (on top of being generally unphysical).

Again you didn't read. He does mention this. The particle will not move when placed upon the spherical dome and the math for the time inversion version replicates the inverse behavior. Whether the whole thing is physical or unphysical is besides the point he is talking about nondeterminism of Newtonian mechanics itself.

>There is just too much wrong in this example, and I suspect you are in way over your head.

Well you suspect wrong. But that's your prerogative. I suspect you're not even really reading the relevant material and just arguing for arguing sake but that's my prerogative. Your judgement makes me suspect you defer to authority, in which I will reply to you that there are many many scholarly papers on the topic of Nortons Dome and There is no definitive consensus among experts on what the dome itself proves about Newtonian Mechanics. Such a self assured stance coming from you literally flies in the face of many experts who have considered the problem far more thoroughly than you or I.

You seem to have in mind some regularization procedure to get past the singularity. I don't agree this is really a valid solution. It wouldn't satisfy the original ODE! (Note that one can regularize e.g. simple collisions of two of the bodies by changing coordinates, see http://www.math.tifr.res.in/~publ/ln/tifr42.pdf.)

I think you're getting too bogged down in trying to think physical/heuristically here. There's a well-defined math problem, a system of ODEs. It's a theorem that, in the case of a collision, there's a solution on some interval [0, t_0), and as the system approaches t_0, the masses collide and at least one coefficient in the equations blows up, rendering it undefined. You cannot give a solution there (in the original coordinates) because the problem isn't even defined at that point.