>So theoretically speaking, they are deterministic, but practically they are unpredictable. The three body problem is REFERING to the ideal case where only a perfect model is considered. Within this model we don't even know the math to calculate it. So, in short, we lack a "deterministic theory" about this problem at all. What's going on here is an assumption. We assume that an idealistic scenario will always produce…
No. We know there is a deterministic solution, even if we can't calculate it.
Scientists find an effective solution for the three-body problem
211–220 of 264 posts
Re: Scientists find an effective solution for the three-body problem
#212Earlier quoted context omitted.
"If we don't know whether a closed form solution exists then how can we even know if it's deterministic. You can prove determinism by finding me a solution that's closed and deterministic." Why would it be necessary to have a closed form solution to know that the solution is deterministic? By the first reference in my previous comment, two identical initial conditions lead to identical time evolutions. That is litera…
https://news.ycombinator.com/item?id=28184786
The example you quote uses Newton's laws to model a system with ODEs that do not satisfy the hypotheses of the uniqueness and existence theorems. It is a nice example, but it is not relevant here, because the ODEs for the three-body problem do satisfy the hypotheses of those theorems. This particular problem is totally deterministic.
"I also said that you CAN prove determinism to me by finding a closed form solution that is deterministic. I never said you needed to nor did I say that was the only way."
I agree. One need not give a closed form solution to show it is deterministic. I gave you another way.
"Your examples are for special cases. Can you prove it to me that for N-bodies and all possible initial conditions that all solutions are deterministic?"
What "special case" is being considered by the term paper I linked above? What are the initial conditions you are worried about, to which the ODE uniqueness and existence theorems do not apply? In other words, where you do think there is a loss of generality?
(More precisely, we have a unique solution up until any collision takes place, if it does at all. But this is not a problem with determinism, it is a problem with modeling the bodies as point masses. We could imagine them as hard spheres and then this issue would disappear.)
Re: Scientists find an effective solution for the three-body problem
#213Here is a link to the arXiv version of the paper: https://arxiv.org/abs/2011.00010 Incidentally, I used to work in dynamics and Hagai is great to work with. To give you a sense of him, I went to a conference on dynamics that he organized about six years ago, and he opened it with the following story: Many years ago, the philosopher Nasreddin was on his farm looking out into the distance and saw some people approachin…
I don't get it. Is it just meant to be funny or is there a deeper message?
IANAP, but it calls to mind "Winger's Friend" which was a response to Schrödinger’s Cat that says:
imagining a (human) friend of his shut in a lab, measuring a quantum system. He argued it was absurd to say his friend exists in a superposition of having seen and not seen a decay unless and until Wigner opens the lab door. [1]
The Ravenous Bugblatter Beast of Traal is a vicious wild animal from the planet of Traal, known for its never-ending hunger and its mind-boggling stupidity. One of the main features of the Beast is that if you can't see it, it assumes it can't see you.
Re: Scientists find an effective solution for the three-body problem
#214Earlier quoted context omitted.
https://news.ycombinator.com/item?id=28184786
The example you quote uses Newton's laws to model a system with ODEs that do not satisfy the hypotheses of the uniqueness and existence theorems. It is a nice example, but it is not relevant here, because the ODEs for the three-body problem do satisfy the hypotheses of those theorems. This particular problem is totally deterministic. "I also said that you CAN prove determinism to me by finding a closed form solution…
>What "special case" is being considered by the term paper I linked above.
Not referring to that. Referring to the Wikipedia version which immediately stated it's a special case. Anyway you're not focusing on that you're talking about the paper and the paragraph at section 2.1.
Your paper assumes the the theorems apply. They don't apply. Why? Because the equations are not Lipschitz continuous when bodies are really really close to each other. They only apply when the bodies are far away from each other which is not always the case for the 3 body scenario. This completely destroys the one solution property and as a result determinism.
Again we are both wrong. I said I we didn't know, you said it's deterministic; but now we do in fact know that it is non-deterministic.
Re: Scientists find an effective solution for the three-body problem
#215Popular belief, and This paper, imply that the three-body problem is unsolvable. The authors mention Poincaré's work demonstrating this.
However, they don't mention Sundman's work in the early 1900's proving that the n-body problem can be solved as a converging power-series.
Sundman's solution is correct, but the series converges very slowly and is impractical.
Since an analytic solution was found over 100 years ago, why are we still debating whether one exists?
Re: Scientists find an effective solution for the three-body problem
#216Earlier quoted context omitted.
You believe that the first clause somehow balances the second? When this (partial) quote is used, it's generally in a context where the first clause is arguably irrelevant. I don't think it's like your "more or less" analogy. There are not many corporations that fail to benefit when the country does well, so the first clause is broadly agreed upon. The second clause, however, is controversial, and has implications th…
> You believe that the first clause somehow balances the second? It doesn't matter if I believe it or not. It's a hatchet job to selectively misquote people to pursue the journalist's agenda.
You think it is selective misquoting. I think it's entirely appropriate quoting. We disagree.
Re: Scientists find an effective solution for the three-body problem
#217Earlier quoted context omitted.
The comfort in shifting your locus of control outward comes from relieving the shame of failure, not from being an overall positive experience. In fact, it's common for people to both take credit for their successes while blaming their failures on external factors to relieve the shame.
Blaming failure on others or external factors doesn't lead to success. It leads to bitterness and resentment.
Right, I wasn't really disputing that.
Re: Scientists find an effective solution for the three-body problem
#218Earlier quoted context omitted.
The example you quote uses Newton's laws to model a system with ODEs that do not satisfy the hypotheses of the uniqueness and existence theorems. It is a nice example, but it is not relevant here, because the ODEs for the three-body problem do satisfy the hypotheses of those theorems. This particular problem is totally deterministic. "I also said that you CAN prove determinism to me by finding a closed form solution…
>What "special case" is being considered by the term paper I linked above. Not referring to that. Referring to the Wikipedia version which immediately stated it's a special case. Anyway you're not focusing on that you're talking about the paper and the paragraph at section 2.1. Your paper assumes the the theorems apply. They don't apply. Why? Because the equations are not Lipschitz continuous when bodies are really r…
"Your paper assumes the the theorems apply. They don't apply. Why? Because the equations are not Lipschitz continuous when bodies are really really close to each other. They only apply when the bodies are far away from each other which is not always the case for the 3 body scenario. This completely destroys the one solution property and as a result determinism."
Take any initial condition with distinct positions for the three masses. There are two cases: the bodies collide, or they don't.
If they collide, there exists a unique solution up until the time of collision.
If they don't collide, there exists a unique solution for all time.
This is essentially proved in the linked term paper. The Lipschitz constant of the coefficients blowing up is what prevents continuation past the point of collision, but this is not a problem until then.
Do you disagree with any of these claims?
If not, do you agree they constitute a proof of deterministic behavior?
Re: Scientists find an effective solution for the three-body problem
#219Earlier quoted context omitted.
We can have these problems simply with classical mechanics’ equations of motion (Newton, Lagrange, Hamilton, whatever). These equations are deterministic, there is no doubt about this. We just don’t have an analytical form. We know that exactly the same initial conditions will lead to the exact same trajectory. What we also know is that the tiniest error will make the trajectories diverge exponentially. It is still d…
After some research I've come to the conclusion that my statement is in fact definitively wrong. What I said was this: We do not know if classical mechanics is deterministic. I am wrong. The correct statement is: We do know that classical mechanics is not deterministic. So essentially I'm wrong but so is everyone else so I'm paying nobody. Source: https://sites.pitt.edu/~jdnorton/Goodies/Dome/ Essentially this is pro…
This felt somewhat wrong to me (as I’m sure it did many people that have take a Physics class) and ended up finding a Reddit discussion about this[1]. Seems like it’s not totally correct because the function for the ball becomes discontinuous
[1] https://www.reddit.com/r/Physics/comments/mn11r/the_dome_a_s...
Re: Scientists find an effective solution for the three-body problem
#220Earlier quoted context omitted.
We can have these problems simply with classical mechanics’ equations of motion (Newton, Lagrange, Hamilton, whatever). These equations are deterministic, there is no doubt about this. We just don’t have an analytical form. We know that exactly the same initial conditions will lead to the exact same trajectory. What we also know is that the tiniest error will make the trajectories diverge exponentially. It is still d…
>We know that exactly the same initial conditions will lead to the exact same trajectory. Do we have a proof of this? Or is it that we just assume this?
Yes we have a proof (in the classical case). It is called the Picard–Lindelöf theorem. The "dome" in your example is called an unstable equilibrium: the ball starts rolling because of an unpredictable perturbation. Small motions are amplified according to the Lyapunov exponent, which in the unstable case is large. Hirsch and Smale and Devaney's book on differential equations and dynamical systems is a good place to learn about this.
In classical mechanics we model a zero-diameter point at the top of the dome, but a real physical dome would be made of molecules vibrating randomly and that starts the ball rolling. At the end of the day, classical mechanics is nonphysical: it doesn't describe nature except as an approximation. Quantum mechanics on the other hand is non-deterministic. Is there a deeper, so-called superdeterministic reality underneath quantum mechanics? Some people think yes, and it has not been disproved, but it is pretty far out of the mainstream from what I can tell.