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>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 i

>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?
    - deltasixeight 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. 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.

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?
  - deltasixeight 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. 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.