5 ms·

https://news.ycombinator.com/item?id=28184786 https://news.ycombinator.com/item?id=28184786

spekcular 5y ago
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.)
- deltasixeight 5y ago
  >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.
  - spekcular 5y ago
    "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?
    - deltasixeight 5y ago
      I cannot agree or disagree with that paper. Unfortunately I don't have time to read the whole thing. BUT, I do have time to read the section you referenced. So I should be more clear with my words: The paragraph on the first part of section 2.1 does not apply for all cases. >If they collide, there exists a unique solution up until the time of collision. If you believe your own words, than it disproves determinism of this mathematical model. Your statement implies there is no unique solution past the point of collision. So you actually agree that that the problem is not deterministic. We've already both arrived at a conclusion here.
      - spekcular 5y ago
        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. (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.) This phenomenon is distinct from the example you gave of the dome with multiple solutions. There, a single initial condition leads to multiple valid solutions, and could reasonably be called non-deterministic. 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. The conclusion you have arrived at is wrong, and I urge you to reconsider. You write: "The paragraph on the first part of section 2.1 does not apply for all cases." What there does not apply to all initial conditions with distinct mass positions?

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.)

deltasixeight 5y ago
>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.
- spekcular 5y ago
  "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?
  - deltasixeight 5y ago
    I cannot agree or disagree with that paper. Unfortunately I don't have time to read the whole thing. BUT, I do have time to read the section you referenced. So I should be more clear with my words: The paragraph on the first part of section 2.1 does not apply for all cases. >If they collide, there exists a unique solution up until the time of collision. If you believe your own words, than it disproves determinism of this mathematical model. Your statement implies there is no unique solution past the point of collision. So you actually agree that that the problem is not deterministic. We've already both arrived at a conclusion here.
    - spekcular 5y ago
      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. (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.) This phenomenon is distinct from the example you gave of the dome with multiple solutions. There, a single initial condition leads to multiple valid solutions, and could reasonably be called non-deterministic. 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. The conclusion you have arrived at is wrong, and I urge you to reconsider. You write: "The paragraph on the first part of section 2.1 does not apply for all cases." What there does not apply to all initial conditions with distinct mass positions?