> It's not that there is no unique solution past the time of collision, it's that there is no solution at all. The model ceases to be well-defined because there is a singularity.
What you meant to say is there is no solution at the singularity. I never said that you didn't say this. I said that your statement implies that past the singularity the solution is nondeterministic.
>For the three body problem to be non-deterministic, one initial condition would have to lead to multiple valid trajectories, which is never the case.
I never concluded this. I concluded that we don't know if this is never the case for the 3 body problem. For classical mechanics in general I concluded that non-determinism is the case.
However let's reiterate what came from your initial post: "You think Newtonian physics is non-deterministic?"
I have proven this statement to not be something that I think, but something that is completely true and thus I have validated what you "thought" was my initial point. That should provide some partial conclusions to your initial points and you have admitted that you agree.
>What there does not apply to all initial conditions with distinct mass positions?
The theory does not apply at the singularities. You stated this yourself, no solution exists.
>The conclusion you have arrived at is wrong, and I urge you to reconsider.
I am in the process of reconsidering by simply speaking to you. That is the purpose of this entire endeavor.
Can you prove to me that past the point of collision the solution is deterministic? Obviously classical mechanics is still highly applicable outside of the the singularity. So after the given initial conditions and up to the point of collision the solution is deterministic.
During the collision the model fails to describe anything.
After the collision there are multiple outcomes that can occur depending on factors you assume and make up in your regularization scheme. All of these possible paths are governed by classical mechanical laws as they are outside of the singularity but we have several possibilities here. Or maybe not. Who knows. My claim is nobody knows. But we do know the system is still ruled by physical laws so the space of possible solutions is bounded. Basically I'm asking you to prove to me that within this space there is one deterministic unique solution.
>As I noted above, this is a residue of the idealization of the problem and could be removed by modeling with hard spheres instead of point masses.
Sure. But that's not the point. If you want to talk about non ideal cases things become weird even when you get really close to the singularity.
> However let's reiterate what came from your initial post: "You think Newtonian physics is non-deterministic?"
> I have proven this statement to not be something that I think, but something that is completely true and thus I have validated what you "thought" was my initial point. That should provide some partial conclusions to your initial points and you have admitted that you agree.
I'm been aware of the dome example for some time. I didn't think such pathologies were worth discussing because they are irrelevant to the current problem, where multiple solutions for the same initial condition do not exist.
>> What there does not apply to all initial conditions with distinct mass positions?
> The theory does not apply at the singularities. You stated this yourself, no solution exists.
You are repeating what I said. Initial conditions with two or more masses at the same position don't have well-defined coefficients, so one can't meaningfully speak about there being a solution. I see no problems with determinism here.
> Can you prove to me that past the point of collision the solution is deterministic?
It doesn't make sense to talk about a solution past the point of collision. One doesn't exist.
> After the collision there are multiple outcomes that can occur depending on factors you assume and make up in your regularization scheme. All of these possible paths are governed by classical mechanical laws as they are outside of the singularity but we have several possibilities here. Or maybe not. Who knows. My claim is nobody knows. But we do know the system is still ruled by physical laws so the space of possible solutions is bounded. Basically I'm asking you to prove to me that within this space there is one deterministic unique solution.
I think this is where the root of your confusion lies. You write "after the collision there are multiple outcomes that can occur depending on factors you assume and and make up in your regularization scheme." The classical three-body problem (discussed in the article and the references I gave) is a problem about a system of differential equations. The formulation is entirely mathematical: I give you some differential equations and the initial conditions, and I want a solution. The physics only enters in the selection of those equations; once I write them down, it's a math problem. Further, it is a theorem that solutions do not exist past the point of collision.
In summary, we know rigorously there is no non-determinism in the three-body problem, because it is a theorem that the same initial condition can never give rise to two distinct solutions.
>I'm been aware of the dome example for some time. I didn't think such pathologies were worth discussing because they are irrelevant to the current problem, where multiple solutions for the same initial condition do not exist.
If you're aware that means you knew your own statement was false and you were attempting to argue a point you didn't believe. The pathology was expanded upon by you. I talked about the n body problem, your statement expanded it to all of newtons laws.
>You are repeating what I said. Initial conditions with two or more masses at the same position don't have well-defined coefficients, so one can't meaningfully speak about there being a solution. I see no problems with determinism here.
Yes I am. Because you asked me a question where the answer involves repeating it. I do see a problem with determinism, because the theorem that is invoked by the paper is about uniqueness. The theorem is called "the existence and uniqueness theorem" and it gives a set of conditions under which an initial value problem has a unique solution.
One of these conditions is Lipschitz continuation which fails at the singularity. Thus the existence and uniqueness of a solution is not guaranteed and thus the theorem doesn't apply. No guarantee of existence means no guarantee of completeness. No guarantee of uniqueness means no guarantee of determinism.
>It doesn't make sense to talk about a solution past the point of collision. One doesn't exist.
This is not true. By definition if the particle is past the point of collision it is not in a singularity and therefore it CAN be modeled by Newtons laws of motion. We just don't know from Newtons laws when it emerges from the singularity or what the exact initial state is outside of the singularity. That being said the state of the particle is still bounded by newtons laws and thus there are certain things we can still say about the particle. For example while this is isn't technically newtons law we know that Energy/mass is conserved before, through and after the singularity.
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.
I have already explained why you are wrong. There is however a new error here:
> 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.
> I have already explained why you are wrong. There is however a new error here:
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:
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.
This has grown too long for me to reply to fully, but I want to make one technical clarification. When I say "singularity," I mean that the coefficients blow up (aren't continuous because they shoot toward infinity in finite time). When author of the pdf file you link to writes "singularity," he means "C^1, but not C^2." You might want to re-read my posts above with this in mind.
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.
>When I say "singularity," I mean that the coefficients blow up (aren't continuous because they shoot toward infinity in finite time). When author of the pdf file you link to writes "singularity," he means "C^1, but not C^2." You might want to re-read my posts above with this in mind.
This is just pedantry. It doesn't matter in what space or what variable the singularity occurs if one exists the solution is undefined at time t=T. This is common knowledge. Additionally in your own words you said there wasn't a singularity and that is definitively false both technically and in terms of common mathematical parlance, there IS a singularity. So this is more of a technical correction rather than a clarification.
>You seem to have in mind some regularization procedure to get past the singularity. I don't agree this is really a valid solution
I don't see it as a valid solution EITHER. regularization is just making shit up. I don't think you understood my number theory example.
Put it this way. With number theory 1/t is undefined at t = 0. But at the exact left side and right side of t = 0 everything is defined again. There is no mathematical property preventing this from happening with numbers. Same with the ODEs. Undefined at the point of collision with no explicit mathematical rule saying it's not defined everywhere else. So really it's [0, t_0) and (t_0, infinite].
Either way there is a singularity at the top of the dome so what occurs on the dome is isomorphic to a system of n-bodies at the singularities. Simply look what happens at t > T for nortons dome. For some arbitary time T the system has an infinite but bounded amount of directions to roll off the dome. So infinty possible T's with infinite possible directions. It can at any time emerge from the singularity and roll in any direction. That is nondeterminism. Not even worth fully deriving the motion mathematically because of so many possibilities.
This is THE SAME thing for the singularity in the n body problem, which is probably why the standard procedure to "solve" this sort of problem past the singularity (and without regularization) is uncommon. At an arbitrary time T the particle may emerge from the singularity with arbitrary unknown properties. But we do know that arbitrary laws still hold like the conservation of mass and energy, we know the particle isn't going to teleport 20 light years away or anything like that. So there are an infinite amount of possible outcomes but those outcomes are bounded by the laws of newton JUST like what happens to the particle on top of Nortons Dome.
I think what's going on here is that you're bogged down by the procedural math routines to solve these types of problems. You have to think a bit outside of the box in order to deal with paradoxes and singularities, yet you must stay within the logical realm of axioms and theorems.
There is a proof that shows a contradiction that basically explains why certain things are undefined at certain points. There is no proof that says the ODE is only defined on an interval up to a singularity. The proof only demonstrates it is undefined AT the singularity.
Every possible state of the particle emerging from the singularity at every possible time t > T is a potential solution.
> Put it this way. With number theory 1/t is undefined at t = 0. But at the exact left side and right side of t = 0 everything is defined again. There is no mathematical property preventing this from happening with numbers. Same with the ODEs. Undefined at the point of collision with no explicit mathematical rule saying it's not defined everywhere else. So really it's [0, t_0) and (t_0, infinite].
Suppose the first collision takes place at t_0, and the initial conditions are x_0. I agree you can give solutions f_1 on [0, t_0) and f_2 on (t_0, infinity) such that the limit of f_1 as t goes to t_0 from the left is equal to the limit of f_2 as t goes to t_0 from the right (at least in the case of double collisions – I'd have to think a bit about the general case).
I do not agree this constitutes a solution of the system of differential equations with initial condition x_0 on [0,infinity). To have a such a solution means, by definition, that there is some function f(t) defined everywhere on [0,infinty) satisfying the differential equation for all times in that interval. As already discussed, that is impossible because at t_0 the coefficients of the ODE are undefined.
You might respond that you're not claiming you have a solution on [0,infinity). (It's a little ambiguous to me.) But "a solution of the three body problem with initial conditions x_0" means exactly a function f(t) defined on some interval [0,A) with f(0)=x_0 satisfying the ODE on that interval. And the only way I can understand your remarks is to take A to be infinity.
(Note this isn't even a determinism vs. indeterminism issue. It's about what is, definitionally, a solution to an ODE.)
(Note also the difference with Norton's dome: in the dome problem, everything –– coefficients of the equation and the solution –– is defined for all positive times.)
What you meant to say is there is no solution at the singularity. I never said that you didn't say this. I said that your statement implies that past the singularity the solution is nondeterministic.
>For the three body problem to be non-deterministic, one initial condition would have to lead to multiple valid trajectories, which is never the case.
I never concluded this. I concluded that we don't know if this is never the case for the 3 body problem. For classical mechanics in general I concluded that non-determinism is the case.
However let's reiterate what came from your initial post: "You think Newtonian physics is non-deterministic?"
I have proven this statement to not be something that I think, but something that is completely true and thus I have validated what you "thought" was my initial point. That should provide some partial conclusions to your initial points and you have admitted that you agree.
>What there does not apply to all initial conditions with distinct mass positions?
The theory does not apply at the singularities. You stated this yourself, no solution exists.
>The conclusion you have arrived at is wrong, and I urge you to reconsider.
I am in the process of reconsidering by simply speaking to you. That is the purpose of this entire endeavor.
Can you prove to me that past the point of collision the solution is deterministic? Obviously classical mechanics is still highly applicable outside of the the singularity. So after the given initial conditions and up to the point of collision the solution is deterministic.
During the collision the model fails to describe anything.
After the collision there are multiple outcomes that can occur depending on factors you assume and make up in your regularization scheme. All of these possible paths are governed by classical mechanical laws as they are outside of the singularity but we have several possibilities here. Or maybe not. Who knows. My claim is nobody knows. But we do know the system is still ruled by physical laws so the space of possible solutions is bounded. Basically I'm asking you to prove to me that within this space there is one deterministic unique solution.
>As I noted above, this is a residue of the idealization of the problem and could be removed by modeling with hard spheres instead of point masses.
Sure. But that's not the point. If you want to talk about non ideal cases things become weird even when you get really close to the singularity.