Scientists find an effective solution for the three-body problem
phys.technion.ac.il
phys.technion.ac.il
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 approaching. He had heard that there were bandits in the area and became afraid that they would come, beat him, and steal all his things. So Nasreddin ran away. As he ran, he came to a graveyard, and there he found an open grave. In case the bandits followed him, he decided to take off all his clothes, get in the grave, and pretend to be dead.
It turned out that the men were not bandits, but were Nasreddin's friends. They saw him run off and wondered where he was going and so followed him. As they were walking through the graveyard, they came upon the open grave and saw Nasreddin lying there naked. So they asked him, "Nasreddin, why are you lying in this grave with all your clothes off?"
Nasreddin opened his eyes and saw it was his friends, and replied, "My friends, there are some questions which have no answers. All I can tell you is that I am here because of you, and you are here because of me."
Wrapped in a funny story.
buy binoculars
For example, imagine a three-body planetary system which exists in a pseudo-stable configuration for millions of years, until suddenly one of the planets gets slung off on a wild orbit and ejected from the system. A layman might be inclined to ask, “whoa, that’s weird, why did that happen?!” while the physicist will reply “All I can say is that planet A is where it is because of planets B & C, and B & C are where they are because of A.”
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.
So theoretically speaking, they are deterministic, but practically they are unpredictable.
I mean, you can certainly still predict to a certain (probably high) level of accuracy, but ultimately that motion is also influenced by factors outside your model.
For a two body problem, you nudge one of the bodies and it is forever off by a small amount, but your predictions into infinity require only a small adjustment to compensate.
For a three body problem you nudge one of the bodies and only for a very short time do your previous predictions stay true, the change amplifies until nothing you thought might happen before the nudge means anything at all, and a common occurrence is one of the bodies being ejected.
So the problem eventually solves itself?
How does this even work? Where is a place in the universe where there are only two bodies?? Where would the nudge come from if not from a third body?! A ghost?
It also doesn't necessarily matter that much if there are more than two bodies, if the gravitational influence of other bodies is small enough, then you can model as a two body problem.
Each planet and the sun can be done like this, ignoring all of the other planets. Each moon and its planet can be considered a two body system ignoring the rest of the moons.
If you just randomly generated a bunch of massive bodies and pressed play, you would have few 2 body systems and a lot of chaos, but that’s a problem that solves itself as the chaos results in either collisions or ejections.
With a three-body problem any slight shift causes a wildly different trajectory, bearing no resemblance to the original so your measurements of the initial condition have to be perfect.
Thanks for mentioning the existence of an analytical solution at all though, I wasn't aware of that.
This is not universally true. Error behavior is a function of the particular problem, the algorithm used to approximate its solution, and the properties of input data. A large subtopic of numerical analysis is concerned with this kind of stuff. See [1] or [2] to get a flavor.
If the three had roughly the same mass would be different story.
The issue with the three-body problem is that it is chaotic, meaning any error will eventually grow to take over the entire solution, making prediction impossible, even in theory. Every chaotic system lacks an analytical solution, but not every system without an analytical solution is chaotic.
X is random with respect to Y, if knowing Y makes no difference to your predicting that X.
QM systems are indeterminate, they are random in the above sense /because/ they are indeterminate. But that isnt what random means.
It's hard to define randomness. I think non-determinism is better than your definition.
Non-determinism is an incoherent definition of randomness; classical physical processes are entirely deterministic.
The point of a coinflip being random is that it is random with respect to the information both observers of the coinflip have. It isnt random with respect to /any/ piece of information.
There are almost no processes which are non-deterministic in this sense. Not enough to bother calling them random; and in physics we do not: the word is indeterminate. Randomness has nothing to do with quantum mechanics; it wasn't invented in the 1920s. It's an epistemic condition.
The RANDOM variable X, st. X ~ N(mean, std) provides a random number x -- x isnt random with repect to the outcome which produced x; nor is it random with respect to an index of a vector in which it is contained.
I would say for a given variable to be random, it must not be predictible, given any other variables that humans can know.
I don't think it makes sense to say that X is random "with respect to Y", that's just the definition of independence.
And a constant variable is independent from all other variables, but it's definitely not random.
I dont think many people have thought enough about the world to appeal to their intuitions.
Randomness isnt quantum indeterminacy.
The n-body problem where you add further bodies, even very small ones, is even harder.
https://physics.stackexchange.com/questions/403574/what-situ...
The actual underlying math is deterministic.
It's maybe similar to predicting the weather - we can have all the perfect equations in the world for fluid dynamics and heat flow etc, but until we have system-invisible temperature and humidity sensors for every square millimetre of atmosphere and earth volume, we won't be able to predict the weather very accurately or very far ahead.
So yes, you could predict the locations computationally to an arbitrary point in the future if you knew their starting locations and velocities with perfect precision; but in practice of course you cannot know anything's position with perfect precision, so your simulation would become inaccurate relatively quickly.
edit to add: and I believe that what this paper discusses is not a solution to the above, but rather a way of getting around it by modeling some types of three-body behavior as if it were truly random, rather than chaotically deterministic.
Measurement devices are designed to be very sensitive to some things and very insensitive to others; for example, you want a clock to be sensitive to how much time has passed but not what the temperature or air pressure are; you want a thermometer to be sensitive to the temperature but not how much time has passed or the air pressure; and you want a barometer to be sensitive to the air pressure but not the temperature or how much time has passed.
It's easy to make a device that's sensitive to all three, like a glass jar partly full of water, upside down in a bowl of water, resting on a bed of gravel in the bottom of the bowl, so that some air is trapped inside the jar. The water level inside the glass jar will go up when the air pressure goes up and down when the air pressure goes down. But it will also go down when the temperature goes down and up when the temperature goes up, because the trapped air will expand and contract. And over time water will evaporate from the bowl, reducing the water level outside the jar, so over time the water level inside the jar will go down.
Usually metrology involves either reducing or eliminating these extra influences (a mercury barometer works the same way as the device described above, but is much less sensitive to temperature because it doesn't have any trapped air; and it's less sensitive to time because mercury evaporates very slowly, and the level of the mercury outside the tube is very low) or balancing them against one another so they precisely cancel out. Chaotic metrology would seem to require a different approach.
Wow, phrased like that it sure sounds obvious, and yet somehow I never thought about it.
For example, the standard way to make an electronic thermometer is by, more or less, measuring the current across a semiconductor diode at a given voltage. This current is an exponential function of the ratio between the voltage and a "threshold voltage" or "thermal voltage" Vt multiplied by an "ideality factor" n: I = Is (exp(V/(nVt)) - 1).
The threshold voltage Vt varies linearly with temperature (it's kT/q, depending only on Boltzmann's constant and the charge of the electron, about 25 mV at room temperature), so in a sense the current at a given voltage is a measurement of the temperature. But the ideality factor n depends on the purity of the semiconductor material (generally in the range 1.0 to 2.0), and the saturation current Is depends on the physical size of the diode junction. Moreover, the ideality factor can change over time as the diode ages. So we're in the position of simultaneously measuring the temperature, the size of the diode, and the quality of its aged semiconductor material.
The solution usually taken, as I understand it, is to measure the current through the same diode at two given voltages, one after the other, and to use a standard value for n which is good enough. Then the ratio of the two voltages tells you nVt (as long as the "- 1" is too small to matter) and from that you can calculate the temperature. In theory, by taking three or more measurements at different points in the I-V curve, you could correct for unknown n as well, but I haven't read of anyone doing this; instead, for high-precision thermometry, they use an RTD.
(Actually, you measure the voltage at two given currents, because that way you don't burn up your temperature sensing diode if the temperature is a little higher than you expected; the power dissipated then varies logarithmically with temperature rather than exponentially. But it comes to the same thing in the calculations.)
It's actually even worse than it sounds, because in fact when you measure a voltage, you're always measuring it with respect to some reference voltage, so your actual measurement is a function of the temperature, the saturation current Is, the ideality factor n, and your reference voltage Vref, which is typically subject to an error of around 2%. But you will note that the ratiometric approach described above cancels out any errors due to Vref, because the ratio of the two voltages will be unaffected by a wrong reference voltage, as long as it's the same wrong reference voltage. So you stick a big capacitor on it and take the measurements in quick succession.
All of this is, from a certain point of view, in the service of making the number you finally produce very sensitive to the temperature of the diode and very insensitive to other factors, like the battery voltage, the temperature of the rest of the thermometer circuit, the age of the components, the humidity in the air, and so on. But all of the actual physical quantities being measured on the diode are the complex mix of factors described above.
MIMO antennas or phased-array receiver antennas or microphones are another example: the signal at each antenna/microphone is a linear superposition of all the differently-phase-shifted source signals, and you process that data to get independent measurements of all the original source signals.
I wouldn't be surprised if chaotic metrology offered new ways to measure very tiny differences, but I suspect it will take a lot of time to figure out the math to make that work.
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.
I really loved the entire trilogy though. Each book had a very different vibe and addressed a completely different topic/problem.
There are so many problems with that book. The inaccurate title is only the tip of the iceberg.
Life is strange. As if God is reminding me something.
It was only yesterday Overthinking [1] was submitted on HN. A little over 10 years ago many of my friends and colleagues told me to stop overthinking about things. It was causing me some stress and depression because when you start doing analysis many level deep the only conclusion is any small variance will simply cause Chaos. I look it up on the internet and that was the first time I learn about three body problem, Chaos theory, and the much more widely known butterfly effect.
I wish I was taught about this in school or told a lot sooner. To me it is much more about life than it is to maths or physics. Where everything could start out as deterministic, and yet the small difference made end results unpredictable. Over time it also evolved or taught me another concept, many many things or solutions in the world are somehow counter-intuitive.
Then I had a few successful project under my belt, but when I was asked in a Job interview I always attribute to "luck" more than anything else. Which happens to be a word HR and many people hate. Americanism ( which also spreads to non-Americans working inside American companies ) views on the world suggest if you work hard you will get it. I wish that was the case, but there were hundreds if not thousands of known moving parts. And possibly thousands of other unknown unknown. It worked. We worked hard. And it worked. It was everyone involved and lots of luck. I was only a small part of it.
I am sure those who interview me are all pretty smart. May be they should try to solve the three body problem.
That said, even though I am not an MMORPG player, I'd be interested in seeing it.
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!
Thanks!
I think a lot of people would have a hard time answering: "why do you play?"
Is it because of deterministic outcome? Would be interested to hear more about the book as well.
America is a big place. I'm seventh generation American, with a patriotic family.
I wasn't raised to believe that if you work hard you WILL get it. No, it's that if you DON'T work hard, you WON'T get it.
You see, success is hard work + luck. You can have luck without hard work, but you have to have a lot more of it to get rich and you still might squander it if you didn't earn it because you won't know what to do with it if you get it by pure chance
You can have hard work without luck, too, like most of the folks in flyover country have. They know they aren't getting rich, they're just trying to get by.
But you can't have real success without both hard work and luck. You might win the lottery with just luck, but you won't wind up running a successful enterprise.
I don't know who is learning from their parents that if you work hard you'll get rich. Mine taught me that if I work hard and have a little luck, I'll get by. A little more luck and I'll be successful. A little less, and I might need to rely on my family or community. That's what they're for.
There's this characiture of American culture and the idea of our meritocracy that I see represented here and in media and it doesn't ring true to me -- I would be interested to know if the people who think luck is the only necessary component for success are Coastal or Flyover, and how much luck they've had
I know for myself, I've needed both work and luck. Without the work, I never would've been in a position to take the opportunities offered by luck.
Depends on how much you start with.
Rich people. It's the result of survivor bias, "I worked hard and got rich, so if you work hard you can get rich too." And they discount the "luck". And it is reinforced by the fact that being born to wealthy, well connected parents is really luck.
Well said. I think the common misconception comes from reducing these wisdoms into aphorisms that are short, but easily misunderstood. Any adult who has lived more than a few years in the real world quickly understands that hard work doesn’t guarantee success, but that success isn’t going to fall in your lap without putting in work.
The online discourse has become particularly bad, with the pendulum swinging between extremes of “You can do anything if you follow your dreams” to the opposite of “Nothing you do matters because it’s all blind luck”.
The latter, cynical mindset has become particularly popular as a way of dismissing or downplaying the success of others. I can’t count how many times I’ve heard people try to attribute Jeff Bezo’s success to that one time he was lucky enough to receive a loan from his family. Yes, it was a lucky break, but it should be obvious that something like receiving a loan from one’s family doesn’t automatically predispose someone to lucking into building a trillion dollar company. Yet there’s a growing contingent of people who want to believe that Jeff Bezos tripped and fell and landed in the founder seat of a successful company by pure luck.
I think the truth is that a lot of people, especially younger people still finding their way, are insecure about their own success or place in life. It can be extremely comforting to surround yourself with explanations that nothing is actually within your control or that others’ success or happiness is the result of randomness. I think this is why we see the oft-repeated trope (on HN especially) that people who post happy photos on social media must actually be secretly sad and miserable behind the scenes: It’s a convenient excuse to downplay the happiness and success of others.
Ignore the extremes. Accept that success isn’t guaranteed. Know that luck is a factor, but it’s not the only factor. Hard work is your lever to maximize the cards you’ve been dealt. We’re all dealt different cards, but it still comes down to your own actions in leveraging the hand you’ve been dealt.
On the other hand, accepting responsibility for results is empowering, because it means one can be successful.
I don't see anything happy about deciding one is a hapless victim of others.
This is not about success, but about being realistic about what you can expect from your life. Everyone is going to be unhappy from time to time and everyone will have downs ; this is just not what you usually see on social media. So when people say this, it's to reduce people's feeling of being inadequate, not to take away success.
If you think about it, it's actually two sides of the same coin: People only see humongous companies and insane salaries, but not the years of hard work that went into getting there. Similarly, they only see happy faces on social media, but not the bad sides that everyone has.
You are likely to have more success if you believe in work. Certainly believing that 100% of outcomes is luck seems like a bad strategy.
At the level of a society then average beliefs matter. I find some less successful countries seem to obsess over the role of external influences, fate, god, and chance.
But earlier
> if I work hard and have a little luck, I'll get by. A little more luck and I'll be successful. A little less, and I might need to rely on my family or community.
This isn't a meritocracy that you are describing
Reminds me of when the CEO of GM said "What's good for General Motors is good for the country." Except he didn't say that. The press did a hatchet job on him by reporting it that way. The actual quote is "what was good for our country was good for General Motors, and vice versa."
https://en.wikipedia.org/wiki/Charles_Erwin_Wilson#General_M...
Just like if someone says "more or less" and the journalist leaves out the "or less".
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 that are quite independent of the first clause.
"It will be sunny today, and tomorrow there will be snow" - if you hear this weather forecast at 13:00 on a sunny day, the first clause is close to information-free, but the second is very striking.
So it is with the quote from the head of GM.
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.
In a way, they’re right. I’d love a 911 GT3, and I could almost certainly get one, if only for a short period of time, and with the benefit of armed robbery.
There’s a lot of things I’ve “chosen” to consider instead of putting everything aside to chase a dream. In more concrete terms, anyone can have anything they want, but what you have to give up for it matters. And that’s something I think my parents and people like them don’t think about. Not everyone has the emotional construction, or even the ability to give up aspects of their life to achieve what they want. I think a large part of “luck” is when the time comes to make a hard decision like that, some fortunate circumstance made swallowing that pill a bit easier.
I read this as the harder you work the more you're able to take advantage of lucky moments. But those lucky moments still need to happen for you to take advantage of them. I think a lot of people don't like to admit that luck had anything to do with it because we have a culture that often suggests that it's luck or work but not some combination. While there are cases on the extreme ends of the spectrum I'm willing to bet that the vast majority are from a combination of hard work and high luck.
Veritasium did a (pretty obvious) simulation that showed those on the top end up having both high luck and hard work.[0] I think this should make sense to most people given how the simulation was run.
I read that line a lot from successful people: "I have achieved X. IF I can achieve it, you can too. Let me explain how. "
There are plenty of born-rich counter examples. New-rich counter examples as well (see Bitcoin millionaires). Frankly, this is just as false as the other one.
Never mind the fact that “working hard” depends quite a lot on the beholder. For example, I would challenge a lot of those self-declared gritty, hard-working ideologists (such as Bezos and quite a few armchair billionaires) to live a year as a minimum-wage single mother in a city.
In any case, there are lots more hard-working poor than hard-working rich, regardless of how you define hardness. So it’s about as valuable as “all the winners played the lottery”, i.e., amusing to say but not really a good way of living.
Anyway, my feeling is that successive people are very good at gaslighting the others to justify their wealth, and that America has a workaholism problem.
Touches on these themes and really made me think about the overlap of chaos and (“macro”?)determinism.
https://onezero.medium.com/anxiety-is-the-dizziness-of-freed...
My two cents: A well thought out design process tries to augment luck with a controlled progress: where you try to move towards your goals in a systematic, more controlled, way so the final outcome is less dependent on luck but more dependent on your ability to properly adjust and execute your design plan. It might be that luck was more important than the process in your case, but that hard to build on, and more importantly, to make any learnings for the future, it still good to analyze how the process could be made better, how you could better take advantage of the lucky circumstances you had.
Exactly, and it also depends on the timescale and precision you're looking for.
It should be obvious that planets in our own solar system aren't showing up at unpredictable locations after a few years, even though our solar system has significantly more than 3 bodies in orbit.
The chaotic behavior in these systems shows up eventually but it's a mistake to think that it's chaotic from the start. We can, and do, predict these systems quite accurately around the starting conditions and time.
The philosophical mistake in the OP's comment is equating a hands-off chaotic system (n-body problem) with a system that has many feedback loops (a person's life). Planets orbiting in space can't take action to change their trajectories. Humans navigating their lives can and do take actions to change their trajectories.
Humans can make moves to correct their own course. Planets cannot. Equating the two is a misunderstanding of personal agency.
Reframe that to recognizing potential and realizing it with great success.
With n-body simulation problems, we don’t actually observe immediate chaotic behavior following small perturbations. In fact, we can readily simulate these systems with considerable accuracy if we want to spend the compute resources. For example, simulating our own solar system with far more than 3 bodies in play can be done with a high degree of accuracy to timescales far beyond our lifetimes.
However, the n-body problem isn’t a good analogy for your sense of personal agency anyway. You aren’t a chunk of rock floating helplessly through space. You are a human being who can take action to influence your own trajectory. You can apply pressure and course correct in a feedback loop, unlike a planet hurling through the solar system.
That doesn’t mean you can influence everything, but it it does mean that it’s wrong to assume that your life is chaotic or that nothing you do matters. (FWIW, The latter feeling is a very classic, and erroneous, thought pattern present in depressive disorders. Correcting that misconception is a core principle of CBT therapy).
You are not a planet hurtling helplessly through space for billions of years. You’re more like a satellite being launched optimistically into the right general area, but it still has to use the limited amount of thruster energy onboard to push itself into the right place. It doesn’t always work exactly as planned, but not using the thrusters at all would assure failure.
That's a misunderstanding. It's working hard on the right things. Working hard digging a hole then filling it up again will never lead to success.
As for luck, the idea is to put yourself in a position where luck can find you. For example, you'll never meet the partner of your dreams by never leaving the house.
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?
the article talks about an effective solution, which would mark a major step forward.
A chaotic system is pretty much a random number generator, and random number generators can spit out the same number (or nearby numbers) twice (otherwise they wouldn't be random).
“Chaotic” usually means that the difference between two similar starting conditions grows without bound the longer you run the simulations forward. But orbits are closed loops; everything about an orbit is periodic. If two orbiting objects start near each other but have different orbital periods, then soon enough they will be far apart from each other. However, if you keep running time forward then they will end up right next to each other again. The distance between them is itself periodic, bounding the total error in a practical sense.
Combine that with the overall stability of our solar system, and you find that most objects tend to stay in particular orbital families for quite some time. Most objects are near the bottoms of deep potential wells, and the forces that can push them out of those wells are quite small. It is only once they are pushed near the boundaries of those wells that rapid changes can begin to happen.
Of course if it were any other way, then there would be nothing left in the asteroid belt by now. Compare that with Saturn’s rings, which simulations suggest will only last another 100k years, give or take a bit. They must be a relatively recent phenomena.
There are, I think still tools that we can build and use even in the face of this type of sensitivity to initial conditions!
But in general there is nothing for a mathematician to do in a library. It is not like you need access to large number of hard to get books. And if you need access to a book, you probably need a lot of time with that book.
That is if you even need books at all.
Even when I studied theoretical math I wouldn't use books at all. Problems tend to be easily formulated. Once I understood the problem I would walk around, lie on the couch, try stuff on the whiteboard or in my notepad, run experiments on Matlab, meet with friends to discuss the problem over coffee or beer and so on.
I don't remember spending time in a library or hearing about anybody spending time in a library.
It was a piece from a mathematician's diary about walking and coming up with proofs. There is something about a changing enviroment and being on the move that's very fascinating to me.
I guess that one mathematician who likes to sit in libraries probably sits there just for sitting there ;)
There seems to be a lot of research on this topic btw.
"To understand any subject, above all, a mathematician SHOULD NOT pick up a book and read it.
It is the worst error!
No, a mathematician needs to look in a book, and to read it backwards. Then, he sees the statement of a theorem. And, well, he goes for a walk. And, above all, he does not look at the book.
He says, "How the hell could I prove this?"
He goes for his walk, he takes two hours ... He comes back and he has thought about how he would have proved it. He looks at the book. The proof is 10 pages long. 99% of the proof, pff, doesn't matter.
Tak!, here's the idea!
But this idea, on paper, it looks the same as everything else that is written. But there is a place, where this little thing is written, that will immediately translate in his brain through a complete change of mental image that will make the proof.
So, this is how we operate. Well, at least some of us. Math is not learned in a book, it cannot be read from a book. There is something active about it, tremendously active.
[...]
It's a personal, individual work."
Just look very closely.
It’s “only” a fan fiction, albeit approved by Liu Cixin.
It nonetheless gives a nice — and as expected — unexpected description of them.
This also reminds me of the QR iteration, where you loose track of the matrix entries very quickly (after 2 or 3 steps into the iteration), but in the end the diagonal does converge to the eigenvalues.
Another interesting 3-body problem is the quarks in a proton or neutron. These can be critically stable with the resulting magnetic field adding more stability. But physics as a field has truly abandoned all mechanical models in favor or purely statistical ones.
https://www.livescience.com/ai-solves-three-body-problem-fas...
While practically true, this isn't technically correct. If you knew the masses, velocities, and location with infinite precision and could perform all operations with infinite precision, (also assuming no external interaction and quantum mechanics doesn't come into it), you could know the state of the system for long time periods. The problem is we can't measure things that accurately.
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 the same result. But we don't actually know because we don't even have a proper model. From a certain angle what this paper is kind of saying is that this assumption is WRONG and that the underlying model of the ideal case of the three body problem IS probabilistic.
I'm not a math/physics guy but the fact that this "theory" involves probability seems sort of like the same initial cop out that came with quantum theory. It's like we can't explain mathematically why a particle behaves this way but it seems to be obeying a probability so let's make probability the basis of the theory! Problem Solved!
Men are by probability more likely to join engineering than women. Because of this probability should we make up a theory called "The fundamental theory of men and women joining engineering" that is described by a probability equation? OR is it better to find an underlying more "deterministic" explanation for why this occurs?
It may sound like I'm denigrating the probabilistic path here, but really theories like this usually only come about when it's basically impossible to come up with a deterministic version. And technically speaking we can never actually know whether the foundations of the universe are probabilistic or deterministic.
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 deterministic.
AFAICT, their random walk idea is not in the trajectories themselves, but in the sampling of the possible trajectories to assign them a statistical weight, a bit like we do commonly in statistical Physics.
Do we have a proof of this? Or is it that we just assume 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.
I've discussed this in other places but the theorem you reference only applies to functions that are Lipschitz continuous. Not all functions have this property globally and the dome and also the three body problem are two examples of things that are not Lipschitz continuous.
Read it carefully. Nortons Dome is saying that the mathematical model described by newtons laws of classical mechanics is in itself 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.
Never made this claim. Not even Einstein made this claim. Simply put, it felt wrong to Einstein simply because probability is a sort of bayesian outlook on the world. It's an admission that we lack knowledge about a system. Such is the case for much of science but not quantum mechanics?
Hence the quote by Einstein: "God does not play dice with the universe." Either way not saying that quantum mechanics is crap and wrong but this is a possibility the mainstream definitely considers in a very speculative philosophical fashion. The quantum model works extraordinarily well but at the same time it's inconsistent with relativity and you gotta admit something is a bit off here when considered from the Bayesian angle.
Superdeterminism isn't classical determinism, it's a specific idea in the interpretation of quantum mechanics. It's based on the idea that not only is it fully determined which box the particle ends up in, but it's also determined which box the experimenter will look in, and those forced choices conspire to make the experimenter think the particle is actually following the Born rule. At least that's my best understanding of it: physics isn't my thing. See:
https://en.wikipedia.org/wiki/Superdeterminism
Besides the people mentioned in that article, I believe Gerard 't Hooft is an adherent. He has a bunch of articles on his site about a possible classical mechanism underneath QM. But, I think most physicists think that is unlikely.
You might like John Baez's article "Struggles with the continuum", which is about various annoying singularities that come up in areas of physics including classical mechanics:
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 proof by contradiction. Classical mechanics is not deterministic.
My answer was, however, better than everyone else's given that my answer was in itself not absolute and encompassed this possibility and the opposing possibility as well.
[1] https://www.reddit.com/r/Physics/comments/mn11r/the_dome_a_s...
Take a look at this (not the same link above I target a single comment):
https://www.reddit.com/r/Physics/comments/mn11r/the_dome_a_s...
I'll have you know you're debating with higher powers here. Some people here are not your typical people who just took a physics class.
Either way intuition without the math is enough to break your brain.
Imagine the opposite scenario you flick a ball up the dome with just the right amount of force that it comes to rest right at the apex. One property of Newtonian mechanics is that motion is time reversible. Meaning that the opposite motion here should be valid. And the opposite motion of the scenario I described is indeed what occurs. The particle is at rest on the dome and arbitrarily just rolls off randomly.
Note that this is caveated on the site with the fact that this intuition doesn't work with hemispheres. Apparently with a hemisphere you can flick the ball with a certain amount of force and it takes infinite time to reach the apex of the dome. So if you time reverse that it means any ball resting on a hemispherical dome can roll off of the apex arbitrarily but it takes an infinite amount of time. The un-determinism only works for the special dome here called "Nortons Dome."
It can happen with a time-dependent force, i.e. not Newtonian dynamics.
Or if a particle can change its velocity without a force being applied to it, i.e. not Newtonian mechanics.
It is more straightforward to see using Lagrangian or Hamiltonian mechanics, which are better suited to this kind of constrained problem, if one is so inclined.
Why? He proved it mathematically without needing to Lift the entire system into a Functor. Everything works fine here.
>A mass perfectly on the top of it will just stay there. It would need a force being applied to it to move at the time T.
You didn't read the site. All of this is addressed and anticipated. This is an unsubstantiated comment where you barely read the article.
There is a section where he addresses the First law. Then after that section he addresses how your intuition can be helped to visualize the legitimacy of this paradox with time reversal. If I tell you about it here, maybe it will make you interested enough that you'll actually read the site before formulating an unsubstantiated comment.
Imagine that your at the rim of the dome and you flick the ball upwards with the perfect amount of force so that the ball rolls up the dome and rests perfectly at the apex for an indefinite amount of time. Now imagine this scenario time reversed. Boom. Newtons laws are time reversible and so is this scenario. If the time reversed scenario is able can intuitively occur then so can the time reversed scenario which is EXACTLY what norton is describing.
What's going on here is that in the mathematics the apex of the dome represents a place where the mathematical functions do not have a property called Lipschitz continuity. When this property is lost, determinism is also lost.
Do note that Norton is not describing reality as we know it. He is describing Newtons Model of reality and the consequences that occur within the model itself. What happens to a marble in the real world rolling down an actual dome is not something he addressing, he is just addressing newtons mathematical model itself.
I have no idea why you bring up functors. Are you thinking of functionals?
Anyway, constrained systems are awkward in Newtonian dynamics, and are much more natural to solve in Lagrangian mechanics, which can avoid some class of errors. Anyway…
> You didn't read the site. All of this is addressed and anticipated. This is an unsubstantiated comment where you barely read the article.
I did, and he does not. The fact is that in Newtonian mechanics, an object at rest cannot start moving without a change in the applied forces. By definition, if it does, then it does not follow Newtonian mechanics. His explanation is thoroughly unconvincing, because on whichever side you place T, the acceleration is discontinuous at T (continuity meaning lim_{t->T+} a = lim_{t->T-} a = a(T) ). At this point, it’s about as well-founded as any random perpetual motion construct.
The whole dome setup is a troll. There is nothing in the principles he mentions that would not work with an ordinary, half-spherical dome, if it did in fact work. His specific dome sounds suspiciously like an artificial setup to get people hung up in irrelevant mathematical details (on top of being generally unphysical).
> Imagine that your at the rim of the dome and you flick the ball upwards with the perfect amount of force so that the ball rolls up the dome and rests perfectly at the apex for an indefinite amount of time. Now imagine this scenario time reversed. Boom. Newtons laws are time reversible and so is this scenario. If the time reversed scenario is able can intuitively occur then so can the time reversed scenario which is EXACTLY what norton is describing.
But that would not happen, because it is non-Newtonian. What he does in fact demonstrate is that a ball with exactly the right energy does not arrive at the apex in a finite time. He got the contradiction right, but sided the wrong way. Besides, he even mentions himself that the ball would not arrive in a finite time, and we are supposed to believe that this trajectory is the time-inversion image of a ball that definitely leaves the apex in a finite time.
There is just too much wrong in this example, and I suspect you are in way over your head.
https://en.wikipedia.org/wiki/Functor. The functor is a generalization of the concept of changing "space". It specifically refers to the mapping between these spaces. For example changing from json to xml, or changing from cartesian coordinates to polar coordinates, euler angles to quaternions or changing from Newtonian mechanics to Lagrangian.
I am saying there's no point in using a functor if the point is already proven. You gain no ground and telling me to change space because of entropy. Things may be easier in the secondary space but because of information entropy there may or may not be a loss of information and this loss definitively leads to less capability of proving anything.
>I did, and he does not. The fact is that in Newtonian mechanics, an object at rest cannot start moving without a change in the applied forces. By definition, if it does, then it does not follow Newtonian mechanics.
Doubtful as you didn't even address his point. You stated your point as if his counterpoint didn't even exist. Either way a discontinuity is undefined but it is 100% legal to talk about the points where it is defined. The limits are legal to address mathematically as there is no math stating that those points are singularities or non-existent or anything like that. You statement of it being unfounded doesn't move your argument in any direction. You need to prove your point or disprove Nortons point.
>But that would not happen, because it is non-Newtonian. What he does in fact demonstrate is that a ball with exactly the right energy does not arrive at the apex in a finite time. He got the contradiction right, but sided the wrong way. Besides, he even mentions himself that the ball would not arrive in a finite time, and we are supposed to believe that this trajectory is the time-inversion image of a ball that definitely leaves the apex in a finite time.
Maybe instead of reading the entire article really quickly and missing the entire point you should read it more carefully. He is talking about the spherical dome. The ball not arriving at finite time is for the perfect hermsphere. For Nortons Dome such an action is 100% possible under newtons model.
>>The whole dome setup is a troll. There is nothing in the principles he mentions that would not work with an ordinary, half-spherical dome, if it did in fact work. His specific dome sounds suspiciously like an artificial setup to get people hung up in irrelevant mathematical details (on top of being generally unphysical).
Again you didn't read. He does mention this. The particle will not move when placed upon the spherical dome and the math for the time inversion version replicates the inverse behavior. Whether the whole thing is physical or unphysical is besides the point he is talking about nondeterminism of Newtonian mechanics itself.
>There is just too much wrong in this example, and I suspect you are in way over your head.
Well you suspect wrong. But that's your prerogative. I suspect you're not even really reading the relevant material and just arguing for arguing sake but that's my prerogative. Your judgement makes me suspect you defer to authority, in which I will reply to you that there are many many scholarly papers on the topic of Nortons Dome and There is no definitive consensus among experts on what the dome itself proves about Newtonian Mechanics. Such a self assured stance coming from you literally flies in the face of many experts who have considered the problem far more thoroughly than you or I.
> Source: https://sites.pitt.edu/~jdnorton/Goodies/Dome/
Nope, according to this analysis, classical mechanics IS deterministic:
https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is...
http://dsbaero.engin.umich.edu/wp-content/uploads/sites/441/...
http://jamesowenweatherall.com/SCPPRG/FletcherSam2010Man_Dom...
https://www.researchgate.net/publication/271399217_The_Norto...
and Nortons Original paper: http://philsci-archive.pitt.edu/8833/4/dome_100711.pdf
Anyway that analysis you posted actually has most of it's point addressed in the initial two papers. Read it.
> Newton’s laws are deterministic, but they’re not complete.
That second paper identifies Lipschitz condition as missing part, and, by the way, it also states in the abstract:
> I do not seek to conclude that these examples are necessarily strong evidence that classical mechanics is not deterministic; rather, I want to emphasize the legitimacy of pragmatic considerations in deciding what legitimately counts as a Newtonian system
That is, the original Newton's principle of determinacy ("The initial positions and velocities of all the particles of a mechanical system uniquely determine all of its motion.") might be proven wrong, but that does not make "classical" (non-relativistic) mechanics indeterministic, only incomplete. To quote Gruff Davies again:
> If we think about particles’ states, and consider higher orders like jounce, snap, crackle and pop. (and all the way to infinity), we can see that the choice of path of unstable particles is fully determined by their values, so this isn’t evidence for indeterminism, it is evidence for incompletion.
We know at the singularity newtons laws are incomplete so in that region you are correct. Prior to the particle entering a singularity newtons laws describe it deterministically so you are still correct.
At some unknown time when the particle exits the singularity Newtons laws still apply but are no longer deterministic, because we do not know what happened in the singularity. We do know the possible states of the particle are still bounded and controlled by newtons laws but within this boundary we are unable to fully determine its unique path if one should exist.
> We do know that Newtonian mechanics is not deterministic at singularity points
then I fully agree with that. However, that may be fixed by either adding additional requirement (e.g. Lipschitz continuity) or just by not considering Newtonian mechanics applicable to those cases - it is known already that Newton's laws do not fully describe the real word (because quantum uncertainty does exist) and the Lebesgue measure of singularity cases is zero anyway.
In case of three-body problem, the singularities are the case of bodies collisions and yes, those cases are not deterministic, but the configuration without collision is known to be fully deterministic.