Scientists find an effective solution for the three-body problem

>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.

Yes, you can. You can amplify small differences due to an additional gravitational field. Then you can run a model with this disturbance included and optimize the parameters such that the difference between the observation and model vanishes. This optimization landscape is not convex however and gets more different "valleys" with time and by knowing which valley you are in you gain information there are several caviats.

If there is noise in the measurement of the system this flattens the curve meaning it is harder to distinguish which valley of the object you are in. If there is noise in system itself this noise will amplified and more and more valleys become possible with time meaning at some point the system state holds almost no information about the system.

However as long as the system runs the thing under measurement should not move as otherwise gets way more difficult and less possible to optimise the difference between the observed and the system behaviour for a certain in parameter under measurement.

In general this approach is not advisable as the chaotic system would also need parts build to enormous precision for that not impact the system more than the signal that influences the system. So it usually better to go with a decently straight forward approach as there are different systems which also amplify small differences but are a lot simpler to work with such as the measurement bridge.

Thank you for the encouragement. I'm glad to have this approach validated.

The good part about being so low key about this project is that "small" - though I'd argue human-to-human scale communication isn't small, it's human scale ;) - feedback like your comment are extremely encouraging. So far, 0 people have not been interested...which is awesome.

I've been fascinated to discover that the process of writing a book is (can be) a lot like the experience of playing an MMORPG. It's a framework that captures your individual achievement, and that achievement is backed up by legitimately hard work an admirable ability to set and achieve our goals.

I could go on-and-on... Cheers!

I've never talked about this project online, so this comment really does mean a great deal to me. See my comment to another reply in this thread: but I have a great deal of excitement about this topic.

I think a lot of people would have a hard time answering: "why do you play?"

When someone describes their comments as having "a little aggression", usually they vastly and egregiously broke the site guidelines. Objects in the mirror are closer than they appear—we all have a tendency to overestimate what the other person is doing by a good 10x and underestimate what we've contributed ourselves.

You posted a huge number of comments in this largely off topic subthread, provoking others repeatedly with swipes and flamebait (by the time we get to https://news.ycombinator.com/item?id=28193336 that's practically all that's left). This is the way that threads degenerate, and I replied to you because you did by far the most to push it in that direction.

It may be true that other users broke the site guidelines also, and if so they shouldn't. But that doesn't mean what they did was equally bad. I'd have to see specific links to know; it's impossible for us to read all the threads thoroughly (https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sor...), and so it's impossible for moderation to be consistent. This is the same reason why cops don't give tickets to all speeders: there are too many of them. Of course it always feels unfair when you're the one pulled over, and it always feels like the mods are against you when you or your side gets scolded.

https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...

https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sor...