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."
Here’s my reading of it: when speaking about dynamical systems with several interdependent moving parts, humans are fond of asking “why” questions and looking for simple narratives in such systems where satisfying answers may not exist.
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.”
I like that! The way I interpreted it when hearing it at the conference he was saying that we were all here at the conference because he had organized it, but he was there because we were all there to talk about our research. But the great thing about a good parable is that there are endless interpretations. :)
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]
I have seen it come up in the Many Worlds school of thought and reminds me of HHTTG's :
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.
When they say motions of three bodies are random and unpredictable, I assume they mean not able to be modeled with a closed form equation? Seems like the motions would still be entirely deterministic — could still predict the locations of the bodies computationally, given your computer computes faster than reality (at least for a reality with only 3 bodies), no?
They would be deterministic. But they are unpredictable. Minuscule fluctuations (coming from influence of some much smaller bodies, that are not in the model, or approximations in calculations) can lead to dramatic differences in the outcome.
So theoretically speaking, they are deterministic, but practically they are unpredictable.
It's got nothing to do with perturbations from additional small bodies, the problem is exactly mathematically calculating the outcome for exactly three bodies within reasonable time frames even if you assume perfect ideal knowledge about the system. It turns out this is excruciatingly hard.
The n-body problem where you add further bodies, even very small ones, is even harder.
Huh? If this is the case, wouldn’t a 2 body problem also be practically impossible to calculate?
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.
The question is what happens with a small perturbation and how does it "grow".
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.
Yes, but but also the problem really is: "given some initial condition space, can we estimate the likelihood of ejection (how many conditions in the condition space eject) in X timeframe"?
> 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.
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?
You do an accounting of the nearby massive objects and their distances from each other, pairs of objects within certain bounds of mass and distance can be treated like there are only those two objects in the universe with the understanding that there will be a small error because of outside influences. It’s an approximation, but in the right circumstances a very good one.
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.
You're taking it too literally. The 'nudge' means a change in initial conditions, and could result e.g. from measurement uncertainty, not necessarily a literal nudge.
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.
The “nudge” can also come up even with exact starting information. When you calculate the next step and round it to the nearest nanometer, you’ve just nudged the system by enough to eventually make your prediction worthless in the 3-body case.
The relative magnitude of interactions is important. The article specifically mentioned triple star system as an example. Solar system is of course many body, but (a) it has already settled into (sort of) stable configuration (b) there is one massive body and everything else revolves around it. Yes, massive bodies like gas giants have noticeable affect on other bodies, but due to the distances they tend to be in the stable territory.
No. A small error in the initial conditions of a 2-body system produces a small error in the result. In case of a 3-body system, a small error will result in drastically different outcomes. The phase space of a 2-body system is nice and smooth, but a 3-body system’s is more like a fractal.
For a two-body problem the discrepancies between your model and reality increase gradually with time, but it's still possible to predict eclipses decades in advance with a precise albeit imperfect measurement of initial conditions.
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.
No that's not it. The two body problem has a closed, analytical solution, the three body one doesn't, so you need to simulate it. It's a fundamentally different approach.
That's not really the issue. The three body problem does have an analytical solution in the form of a power series, but the problem is that it converges so slowly to be of any practical use.
It is though, if you don’t have a closed form solution you need to use an iterative process to calculate the positions, meaning errors will accrue over time. For a closed form solution that wouldn’t be the case.
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.
There are iterative methods/systems that stabilize over time. For example, symplectic integrators on tame problems oscillate lightly around the true energy of the system over time. The issue here is the properties of the underlying problem, not the set of solution methods.
For all practical purposes eclipses are two two-body problems ... Sun vs Earth-Moon pair, and Earth vs Moon. The scale of the Sun is so vastly larger and distances so great from sun to earth there's no need to do 3-body.
If the three had roughly the same mass would be different story.
Better stated, the 2-body problem can be solved with a finite number of standard operations, i.e. a closed-form expression. This solution does not exist for the 3-body problem.
In principle - yes. Except that we can change our frame of reference, and treat the two body problem as a pseudo one body problem (the lab frame becomes the center of mass of one of the bodies). One cannot do this for the three body problem, which gives us at best a pseudo two body problem.
No, the fundamental difference is that the two-body problem can be solved analytically, you can write down a formula. For three bodies and up you only have numerical solutions, simulations, and they will break down over time.
The problem is that a three-body system is inherently unstable/chaotic. So even if you run a numerical simulation with the same granularity for a two-body and a three-body system, the three-body simulation will degrade much faster than the two-body system. This is unrelated to the fact that there is a closed form solution, there are many stable, non-chaotic dynamical systems that don't have a closed form solution.
The need for numerical simulations is completely unrelated to the predictability and/or stability of the system. If you have a Lyapunov exponent smaller than 0 everywhere, errors don't accumulate and you can simulate numerically for as long as you like.
It's not unrelated, but the implication only goes in one direction. If you have a closed form solution, that implies that you can model it completely forever, but sure the opposite is not true.
Of course. My point was just that you can evaluate a recurrence relation exactly (i.e. with zero numerical error) and still get chaotic behavior. OP’s mistaken point was that the three body problem’s chaos arises solely from numerical error during simulation, which is untrue.
That is not the issue. There are many problems that do not have analytical solutions, but can be approximated numerically to any precision you would like.
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.
It isn't hard to define randomness. It's an epistemic condition on the knowability of Y given X.
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.
The motion isn’t deterministic if free will exists. Launching a rocket into space decreases earth’s rotation speed ever so slightly, which will have a small impact on the moon’s trajectory due to tidal interactions, and so on.
Edward Lorenz summarized chaotic behavior as: "When the present determines the future, but the approximate present does not approximately determine the future."
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.
While we can simulate three (or more) bodies' gravitational interaction, the chaoticness means that any error in initial state, no matter how small, will be hugely amplified. This makes long-term predictions untractable
As I understand it, "random and unpredictable" means impossible to measure the initial state in sufficient precision and/or computationally impossible to calculate in a reasonable amount of time.
It seems the problem is often mis-stated - there is no calculation problem, the problem is in adequately defining/sampling the initial data.
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.
Chaos is the extreme dependence on initial conditions. The three body problem is significantly more chaotic than a two body problem. Welcome to chaos theory!
The initial conditions are usually known up to a small epsilon in practice. I guess this initial error grows exponentially with time, hence "unpredictability".
You can only do that with some error, e.g. your simulation needs some time step that controls the tradeoff between computational resources used and accuracy of the results. For any fixed time step, after enough time your simulation will completely diverge from reality.
On top of theoretically computable, but highly divergent / unstable functions mentioned nearby, there are fully deterministic (non-random, pure) but non-computable functions. The simplest example is a function that answers whether a given Turing machine would stop.
If you freeze each point in time , isn’t the next minute step deterministic? Every time you freeze, you would have all the motion vectors to calculate the next moment, if you expand that using a lot of computation power, ca you solve it that way?
there is no closed form, like you say, and the additional complexity is around ergodicity, i.e. solutions that start close to each other might end up very far from each other after a certain point. this is also an issue with computer simulations as the error might accumulate and push solutions away. in practice, given the amount of cosmological computations people do on a daily basis, including those for satellites and rockets, this might not necessarily be that big of an issue, but i don't work with that stuff on a daily basis.
You sure about that? The fourth body, being so small, can't really effect the motion of the other three; however if you have the evolution of the first three then you can determine the motion of the fourth.
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.
It's not the only possible way to do things, and sometimes it's infeasible, but it definitely makes things simpler. In theory if you have N quantities, and N measurements that all depend on all N of those quantities in known but different ways, you can usually compute the N quantities precisely from the N measurements.
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.
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.
Not really because they're already so chaotic you couldn't be sure what divergences from simulation were inherent and which were due to external perturbation.
Not sure what the discussion is about, but the restricted approximation only works for "insignificant" masses (e.g. a moon if talking about two big planets and a star). And even then for "short" time periods (a few million years). In larger scales even that mass will (I assume you've heard of the butterfly effect) play role.
It's been a while since I've read it, but I seem to recall that there were other planets, but they either got ejected from the system, or fell into one of the stars.
They're gonna feel so silly when they get all the way here to kill us and realize we solved their silly little problem and they have to turn right back around and go home. The looks on their translucent non-existent face things...
Not to say all the other signs of earth those seamonkey tea bag aliens craving for a better future missed with their superior technologies in Alpha Centauris immediate neighborhood.
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.
> Americanism ( which also spreads to non-Americans working inside American companies ) views on the world suggest if you work hard you will get it.
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.
> I don't know who is learning from their parents that if you work hard you'll get rich
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.
> 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.
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.
> 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
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.
> 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.
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.
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.
There's a saying: the harder I work, the luckier I get.
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.
Interestingly though, when asked Americans rate luck as much less important in financial success than other cultures. Euro countries in particular are more comfortable attributing a higher fraction of their success to luck than those in the US. And by asked I mean a reasonable well designed study… I can’t find the ref just now but will post if I do when home again.
It makes perfect sense to have an irrational belief in hard work.
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.
This is wonderfully North American. I would judge countries like France, Germany much more successful than the US because they still have functioning political and cultural life. Those are only things that can survive if individual financial success is not the only metric used.
> There's this characiture of American culture and the idea of our meritocracy
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.
Sure it is. Hard work weights probability in your favour and gives you more opportunities. Surely that's the best anyone can ever hope for? Do you have a system in mind that makes guarantees?
A meritocracy isn't about guarantees, other than ranking is by performance not externalities. By most definitions, a meritocracy is impractical, granted. In a real-world sense, the socio-economic status of every individual on earth is primarily governed by luck/circumstance. Not to say there aren't exceptions, but there has to be an inordinate amount of "merit" AND luck to overcome the initial state, statistically. To whit, a human lifetime is more complex and complicated to navigate, than the 3 body problem.
> No, it's that if you DON'T work hard, you WON'T get it.
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.
> 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.
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."
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 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.
My parents’ attitudes are very much more oriented toward hard work becoming success and that if you haven’t gotten what you wanted, it’s because you haven’t tried hard enough yet.
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.
> 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.
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.
To me it is not surprising that an interviewer would not be satisfied if you attribute your success to luck alone. What can you contribute to a new work place if you rely purely on luck? It is important to identify how you were in a position to take advantage of the lucky circumstances that you had...
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.
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.
Ugh I think back to my server-first raid boss kills in wow, my giant space battles in EVE Online, my lasting relationship trauma from being "catfished" before it was even a word on Second Life, and wish I'd just gone to college or something instead of spending my teens and 20s playing MMOs. What a terrible trap these things are.
Not yet, but I'll start thinking about it. I've never talked to anyone about it other than close friends, I probably should share more as progress continues.
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.
See my reply to a sibling comment, but long story short: thanks for sharing your interest. You are on a (very short) list of individuals that I'll include when I start communicating or publishing some of my progress.
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?"
Lots of classes function are infeasible to compute, but they're deterministic nevertheless. The whole universe might not have enough computational power to give you the answers.
> Lots of classes function are infeasible to compute, but they're deterministic nevertheless.
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.
Weren't there already several solutions known? Wikipedia seems to think so. I find it hard to see if there's anything here that makes this different from the several cases mentioned on wikipedia. The linked paper was too dense for me, sadly...
How good are we at predicting three body movement today with our computers? This is the question I could never find an answer too. Can we do it real time, or few years into the future? Can we do it accurately, to an arbitrary precision? Or is it always fuzzy with statistical outcomes?
We can do it accurately, to any precision you care to pay for. Since n-body gravitation is a chaotic system, getting more precise predictions requires more precise measurements of the current state of the solar system. When it’s not possible to measure things more precisely, we instead run many simulations with small random perturbations in the current state, then classify the simulations to get probabilities.
One way to deal with these kinds of issues is to express your initial conditions as intervals (or even distributions) in the sense that you include the a range of possible values, rather than the most probable one (which is normally implicitly done). So if you measure the earth to be (6±1)e24 kg, then you work with something that looks like [5,7]e24 kg i.e. the segment of the number line corresponding to the possible physical realities that led to your measurement. You'll get a range of different outcomes in the end, and their relative probabilities given your priors. You can do this exactly for some systems, but usually you'll do some monte carlo and hope it's valid. This is similar to classical (linear) error propagation where you carry around an "uncertainty", but chaotic systems don't generally allow you to make the assumption of narrow Gaussians used there.
The system is chaotic so there is a strong dependence on the initial conditions. I suppose that if you don't know these precisely, then at some point even the best computer simulation can't help you much.
It is a chaotic system. Arbitrary small deviations in the initial conditions will result in completely different outcomes. So your simulation will eventually diverge from reality as you cannot measure the initial conditions exactly.
I do accept what you have said (indeed it's chaos theory dogma that I am mostly still on board with!) but I also think reality is quite far from binary in whether chaos-theory concepts apply-to-it-or-not, seems a bit like it's mixed-in everywhere a bit, depending how information is mixing. (I'm calling chaos Class-3 Automata style)
There are, I think still tools that we can build and use even in the face of this type of sensitivity to initial conditions!
Nitpick + a little more thought: Isn't it more correct to say, that initially slightly different conditions might (instead of "will") result in a very much different outcome? Does chaotic mean, that two states which only differ a little must result in vastly different outcomes? I wonder whether there could be states, which are very similar and some condition drives them to converge again. Or is such a thing impossible?
Yes, though frequently the solutions are very similar to each other. For example, if you plot the future of an asteroid in 10,000 different simulations, you’ll probably find that in most of them the asteroid remains in the asteroid belt where it started from, but perhaps in 10% of them it is perturbed enough by Jupiter that its orbit becomes a Trojan, or some other variety. If you look at the details of the 90% where it stays in the asteroid belt, you find that while they are all in different orbits from each other, the differences are not very significant. Just 9,000 rather similar orbits inside the asteroid belt.
“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.
Often yes. How often happens in practice will usually depend on the size of the solution space.
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).
Not to answer your question, but you may be interested to know that chaotic systems can often be effectively controlled by small perturbations: https://en.wikipedia.org/wiki/Control_of_chaos
The photo of Professor Hagai Perets (Left) and Ph.D. student Yonadav Barry Ginat seems to have them in their native environment: the university's Science and Math library. If you zoom you'll see the Math and Science topics listed on the card for the book shelves behind them.
I am sure that there is at least one mathematician who likes to sit in libraries.
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.
I was looking around but couldn't find the piece my prof sent me 5 years ago.
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 ;)
Alain Connes, Fields medalist, talks about going on walks while reading math books in a particular way (and on how a mathematician works and should read a book) [0]:
"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.
Walking helps thinking, that is a well known fact, which I also learned from a mathematician who kept sharing in class how many problems he solved while walking his dog. He would always start his phrases with "while I was walking my dog I realized ...".
There seems to be a lot of research on this topic btw.
I had one famous professor, Gian-Carlo Rota, whose office was covered in stacks of books and journals; I believe that he was an editor of an AMS publication at the time. The next year I had a professor for a class in non-commutative ring theory (I sadly can't remember his name off the top of my head; I do remember that he was a pleasant and brilliant person.), his office was like a monk's room. There was a desk upon which rested a single sheet of paper with a yellow wooden pencil.
I always imagined them as some kind of "plants", given how they're described to replicate/combine and how they're dried up and stored and later rehydrated...
Coming from a non-physics background, I would like to know how would this help us in predicting the future trajectory of a three-body system? How does it improve over the current solution techniques?
While the tone sounds like
a university webpage advertising its scholars' results,
this does seem to be an interesting viewpoint.
(Unless similar viewpoints have been proposed before.)
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.
I'd like to see this extended slightly to give a half-life based on the masses or similar.
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.
> one cannot simply specify the system evolution over long time-scales.
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.
If you know the initial wave function (which you can’t measure, but you could in principal choose and set up three bodies accordingly), then quantum mechanics is deterministic. If you throw the standard model in, you get a mess, though.
>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 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.
"You are completely and utterly wrong and you don't know what you're talking about" is not "just discussion", it's obvious flamewar. Posting like that to HN will get you banned, because it violates what the site is supposed to be for, namely curious conversation. If you don't want to be banned on HN, please follow the rules and don't post like this again.
Well he posted an unsubstantiated comment. I Had a long post and he dismissed it with one line. So there's a little aggression in my response, but I told the truth, did not insult him or do anything of the sort.
There is obviously Zero flame war going on as it's a single post and the rest of the thread extends into a long and "namely curious" discussion which you essentially just hidden from view.
This is selective enforcement and you can totally ban me if you want but if you do I would say you're being unfair, dismissive and a bad moderator. Yeah ban me for that minor thing, but please explain to me how you just completely ignored the person who I responded to.
Additionally a person responding to the post you flagged decided to call me irrational out of nowhere. I flagged, but again ignored. How is this fair at all?
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.
I haven't checked but I think existence and uniqueness for the classical three body problem likely follows from standard theorems on ODEs. If you can articulate why this is not the case, please do.
In fact, more is known: the solution exists and is analytic. From Wikipedia: "However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in the form of a power series in terms of powers of t1/3. This series converges for all real t, except for initial conditions corresponding to zero angular momentum. In practice, the latter restriction is insignificant since initial conditions with zero angular momentum are rare, having Lebesgue measure zero. "
For the three body problem we don't know the closed form solution under Newtonian physics. So if we don't know it, then a closed solution may not even exist. 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.
My answer is the same as stated: we don't know, and that the claim that we do know, is completely wrong.
>In fact, more is known: the solution exists and is analytic
These are special cases, under special conditions. A general solution is not actually known. We don't know. We are only assuming that it's deterministic. Also the solution you provided is analytic not closed.
> So if we don't know it, then a closed solution may not even exist. If we don't know whether a closed form solution exists then how can we even know if it's deterministic.
We can prove that something has some properties even if we don’t know everything about it. Newton’s second law in a many-body system is (in the case of Newtonian gravity) basically a system of independent second-order ODEs. We know a thing or two about these beasts.
Anyway, let me try something (without LaTeX, so I apologise in advance for the notation).
Let’s think about a many-body system (following Newtonian mechanics, and interacting via Newtonian gravity) from the point of view of its positions-velocities phase space.
The solution is not deterministic if there are points (i.e., sets of positions and velocities for all the particles) that are part of at least two trajectories.
Let’s assume we are looking at such a point. The forces F and all their derivatives dF/dr are the same along all trajectories, because they depend only on the positions (which are identical). The velocities and the positions are the same because it is the same point in the phase space.
The third derivatives of the positions w.r.t time are also identical (d^3r/dt^3 = 1/m dF/dt, and dF/dt = dF/dr dr/dt, and the dr/dt are identical).
You can go on to the n-th derivative, n being arbitrary large (d^4 r/dt^4 is a function of only dF/dr, dr/dt, d^2 r/dt^2 and d^2 F/dr^2, all of which are identical along both trajectories, etc).
So, it means that both trajectories have the same position in the phase space, and all their derivatives are identical, i.e., both trajectories are identical. So it is deterministic (there is no point in the phase space that belongs to more than one trajectory).
"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 literally the definition of determinism.
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.)
>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.
"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?
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.
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?
> 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.
> 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.
I'm been aware of the dome example for some time. I didn't think such pathologies were worth discussing because they are irrelevant to the current problem, where multiple solutions for the same initial condition do not exist.
>> 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.
You are repeating what I said. Initial conditions with two or more masses at the same position don't have well-defined coefficients, so one can't meaningfully speak about there being a solution. I see no problems with determinism here.
> Can you prove to me that past the point of collision the solution is deterministic?
It doesn't make sense to talk about a solution past the point of collision. One doesn't exist.
> 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.
I think this is where the root of your confusion lies. You write "after the collision there are multiple outcomes that can occur depending on factors you assume and and make up in your regularization scheme." The classical three-body problem (discussed in the article and the references I gave) is a problem about a system of differential equations. The formulation is entirely mathematical: I give you some differential equations and the initial conditions, and I want a solution. The physics only enters in the selection of those equations; once I write them down, it's a math problem. Further, it is a theorem that solutions do not exist past the point of collision.
In summary, we know rigorously there is no non-determinism in the three-body problem, because it is a theorem that the same initial condition can never give rise to two distinct solutions.
>I'm been aware of the dome example for some time. I didn't think such pathologies were worth discussing because they are irrelevant to the current problem, where multiple solutions for the same initial condition do not exist.
If you're aware that means you knew your own statement was false and you were attempting to argue a point you didn't believe. The pathology was expanded upon by you. I talked about the n body problem, your statement expanded it to all of newtons laws.
>You are repeating what I said. Initial conditions with two or more masses at the same position don't have well-defined coefficients, so one can't meaningfully speak about there being a solution. I see no problems with determinism here.
Yes I am. Because you asked me a question where the answer involves repeating it. I do see a problem with determinism, because the theorem that is invoked by the paper is about uniqueness. The theorem is called "the existence and uniqueness theorem" and it gives a set of conditions under which an initial value problem has a unique solution.
One of these conditions is Lipschitz continuation which fails at the singularity. Thus the existence and uniqueness of a solution is not guaranteed and thus the theorem doesn't apply. No guarantee of existence means no guarantee of completeness. No guarantee of uniqueness means no guarantee of determinism.
>It doesn't make sense to talk about a solution past the point of collision. One doesn't exist.
This is not true. By definition if the particle is past the point of collision it is not in a singularity and therefore it CAN be modeled by Newtons laws of motion. We just don't know from Newtons laws when it emerges from the singularity or what the exact initial state is outside of the singularity. That being said the state of the particle is still bounded by newtons laws and thus there are certain things we can still say about the particle. For example while this is isn't technically newtons law we know that Energy/mass is conserved before, through and after the singularity.
There is no confusion here. A singularity exists at the top of Nortons Dome and Norton is exactly describing the behavior of the particle AFTER it exits the singularity. The exact same phenomena is occurring in the n-body problem as nortons dome.
I have already explained why you are wrong. There is however a new error here:
> There is no confusion here. A singularity exists at the top of Nortons Dome and Norton is exactly describing the behavior of the particle AFTER it exits the singularity. The exact same phenomena is occurring in the n-body problem as nortons dome.
There's no singularity in Norton's dome. The equation d^2r/dt^2 = r^{1/2} is well-defined for all non-negative times t. This is not the case in the three-body problem, where a coefficient blows up in finite time when two bodies collide. The dome example is not relevant here and you are confusing yourself by conflating the two phenomena.
>> >It doesn't make sense to talk about a solution past the point of collision. One doesn't exist.
> This is not true.
And just to repeat myself: it is true, it's a mathematical theorem, you can't continue the solution past the point of the collision. Please check the references.
If you want a heuristic reason for this, note that conservation of energy implies infinite velocities at the point of collision. The situation is inherently unphysical due to it being a mathematical idealization.
edit: If you're serious about the $1k thing, I propose the following. We agree on a precise mathematical formulation of the question "Is the three-body problem deterministic?," an expert, and a charity. We email the expert for comment. If I am right, you donate to the charity. If I am wrong, I will admit to it here, and perhaps donate some money myself. We can put the agreement on our personal websites and post it to HN to publicly commit.
> I have already explained why you are wrong. There is however a new error here:
And I have countered your argument. I get it. You're trying to say that we're talking past each other here and that you're just repeating yourself. No. I addressed your point, you don't need to repeat yourself you only need to counter the new point.
That being said, there is no new error. See this paper here which talks about the singularities at the top of the dome:
Norton is 100% dealing with a singularity and therefore he is describing the exact same phenomenon of what happens to the particle after it exits the singularity.
>And just to repeat myself: it is true, it's a mathematical theorem, you can't continue the solution past the point of the collision. Please check the references.
I never said the mathematical theorem is not true. I said it doesn't apply. I did check the references. The theorem only applies for certain conditions and those conditions aren't met, so the theorem has nothing to say about the system at that point.
Let's be clear. The theorem says one thing and one thing only that a unique solution exists if the conditions are met. When the conditions are not met (at the singularity) the theorem says absolutely nothing and therefore it does not apply. The theorem does not say that the solution does not exist, it just doesn't apply.
>If you want a heuristic reason for this, note that conservation of energy implies infinite velocities at the point of collision. The situation is inherently unphysical due to it being a mathematical idealization.
Doesn't matter. Infinite velocities is something you can't consider because the model is undefined at the singularity but it is defined both before and after the singularity. To give you a heursitic reason as well: imagine the successor function for peano arithmetic in number theory.
We have a number X where X = 0 and subsequent numbers are defined as S(X) or S(S(X))) and so on.
If X = 0 exists and A is any number, then A/S(X) still exists even though A/X doesn't exist (division by zero). This is what's going on here. Energy is conserved at the singularity (X = 0) but what that implies is undefined (1/X). But outside of the singularity the implication 1/S(X) is defined. There's no rule saying that 1/S(X) can't be defined just because 1/X is not defined. We are still in the purely mathematical realm here, we are not talking about physics in reality.
Where the mechanics becomes nondeterministic is that (to continue with the heuristic analogy from above) 1/S(X) where X=0 is nondeterministic. Just like in number theory The literal math states that the system is undefined at the singularity. In other words the math DEFINES the solution to be UNDEFINED at the singularity. However the math itself makes zero statement about the system AFTER the particle exits the singularity. Hence we can say certain things about the particle if/when it exits the singularity even though many things are unknown. This is the reason why Norton is able to arrive at his conclusion of bounded nondeterminism. Certain things can be derived but newtons laws about the particle at 1/S(X) but not enough to arrive at a unique nondeterministic solution. The exact same thing applies to the n-body problem as it does to Nortons Dome.
Also take a look at the paper I linked above and read the "Space Invaders" section. It addresses another possibility of in-determinism in a Newtonian n-body system based on how we arbitrarily choose to define it. This topic alone will initiate another angle in which to approach the question of determinism of an N-body system.
>edit: If you're serious about the $1k thing, I propose the following. We agree on a precise mathematical formulation of the question "Is the three-body problem deterministic?," an expert, and a charity. We email the expert for comment. If I am right, you donate to the charity. If I am wrong, I will admit to it here, and perhaps donate some money myself. We can put the agreement on our personal websites and post it to HN to publicly commit.
I am serious. The problem with experts on this topic is they all have different opinions on the topic. What if I pick the expert and author of the paper I linked above? There is no consensus among experts. I'll think about it but I don't want to be in a situation where the expert picks a solution and I remain unconvinced only to later find another expert who has the opposite opinion.
I will acknowledge that there's an obvious $1000 bias on my side here and that a 3rd party levels the playing ground but unfortunately there is heavy bias on the "expert" side as well. I mean aren't you an "expert"? Hence your confidence on this topic.
I think for now, unfortunately, we're going to have to settle with you convincing me and you'll have to take my word that if you can convince me I'll concede and try to look past my biases. Either way I'll think about your proposal.
Also you should note my initial bet post was flagged so this entire thread is basically dead except for a couple people still responding to me.
This has grown too long for me to reply to fully, but I want to make one technical clarification. 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.
You seem to have in mind some regularization procedure to get past the singularity. I don't agree this is really a valid solution. It wouldn't satisfy the original ODE! (Note that one can regularize e.g. simple collisions of two of the bodies by changing coordinates, see http://www.math.tifr.res.in/~publ/ln/tifr42.pdf.)
I think you're getting too bogged down in trying to think physical/heuristically here. There's a well-defined math problem, a system of ODEs. It's a theorem that, in the case of a collision, there's a solution on some interval [0, t_0), and as the system approaches t_0, the masses collide and at least one coefficient in the equations blows up, rendering it undefined. You cannot give a solution there (in the original coordinates) because the problem isn't even defined at that point.
>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.
> 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].
Suppose the first collision takes place at t_0, and the initial conditions are x_0. I agree you can give solutions f_1 on [0, t_0) and f_2 on (t_0, infinity) such that the limit of f_1 as t goes to t_0 from the left is equal to the limit of f_2 as t goes to t_0 from the right (at least in the case of double collisions – I'd have to think a bit about the general case).
I do not agree this constitutes a solution of the system of differential equations with initial condition x_0 on [0,infinity). To have a such a solution means, by definition, that there is some function f(t) defined everywhere on [0,infinty) satisfying the differential equation for all times in that interval. As already discussed, that is impossible because at t_0 the coefficients of the ODE are undefined.
You might respond that you're not claiming you have a solution on [0,infinity). (It's a little ambiguous to me.) But "a solution of the three body problem with initial conditions x_0" means exactly a function f(t) defined on some interval [0,A) with f(0)=x_0 satisfying the ODE on that interval. And the only way I can understand your remarks is to take A to be infinity.
(Note this isn't even a determinism vs. indeterminism issue. It's about what is, definitionally, a solution to an ODE.)
(Note also the difference with Norton's dome: in the dome problem, everything –– coefficients of the equation and the solution –– is defined for all positive times.)
No look at my wording more carefully. You are misinterpreting it. I said we don't know if a "closed form solution" is deterministic because we don't even know if there is a "closed form solution."
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.
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?
Unfortunately your aggressiveness and rudeness make your comment of lower quality than the one you are answering - read https://news.ycombinator.com/newsguidelines.html to see what site rules you are breaking as well. Betting doesn't make your argument truer, it only points to your own irrationality.
Disclaimer: I am no mathematician nor physicist.
However the intuition is that any set of equations must be deterministic, unless they refer to a non-deterministic function like random(x).
Should you wish to argue that the three body problem can generate true randomness, I think it is your responsibility to refer to links that support your argument. I would expect an argument to refer to mathematical concepts and have nothing to do with the three body problem in particular. I suspect it verges on tautological that if you can define equations then it is deterministic.
>Unfortunately your aggressiveness and rudeness make your comment of lower quality than the one you are answering - read https://news.ycombinator.com/newsguidelines.html to see what site rules you are breaking as well. Betting doesn't make your argument truer, it only points to your own irrationality.
Did you not notice the person who replied to me literally just commented a single sentence and left it at that? It's called an unsubstantiated comment and it is also addressed int the rules you link here: https://news.ycombinator.com/newsguidelines.html
I simply made a bet. There's no aggression here. I like to back up my claims with more seriousness rather then single line comments. Also claiming my post is rude and aggressive when it isn't is in itself rude. I just make claims and I offer to back up my claim with money. Why? I would offer a proof but no proof is known because my answer is: We don't know.
>Should you wish to argue that the three body problem can generate true randomness, I think it is your responsibility to refer to links that support your argument
Except this is not my wish and I never stated such. My statement is that we don't know if it's deterministic. My offer is that I put money on the fact we don't have any definitive proof we do know.
>However the intuition is that any set of equations must be deterministic,
Intuition is just hand waving. There is nothing definitive here I'm sorry. My offer stands to the first person that can give me a proof, $1000. That includes you even though I found the first paragraph of your post dishonest and rude. You called me irrational, that's practically an insult and a flagrant violation of the rules. I did no such violation.
What's going on here is not that I'm violating the rules. But that you're biased. You disagree with me so your bias senses aggression and violations of rules on my side even though I did no such thing. In fact your bias blinds you to your own violation. You called me irrational. Might as well call me stupid. Same personal insult just disguised with smarter wording.
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 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.
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.
I read them. The first paper you referenced do not address any points (it just gives another example, similar to the Norton's Dome), and the second one actually confirms the major point given by Gruff Davies:
> 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.
Incompleteness implies indeterminacy. Davies point is sort of pedantic but the math from Nortons paper is nondeterministic BECAUSE of incompleteness.
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.
The dome is not a proof. It has several flaws and the reasoning is invalid. 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.
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.
>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.
> Why? He proved it mathematically without needing to Lift the entire system into a Functor. Everything works fine here.
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.
>I have no idea why you bring up functors. Are you thinking of functionals?
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.
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
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."
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.
No read the dome example again. You are wrong. Nortons dome is literally demonstrating there is proof for the non deterministic case.
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.
Oh I see what you mean, the dome is a little bit pointy at the top, so the first derivative is not continuous. I missed that at first. Sorry. Still though, this seems like a typical situation in physics where you get superposed solutions to a DE and then throw out the nonphysical ones. Someone else just created a new thread about the dome problem, and web search shows a vast literature about it (like there is for the Monty Hall problem). I guess you're already familiar with the literature so there's not much I can add.
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:
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:
That’s not what I see. I see people, many of whom are not experts in math or physics, discussing the topic. There are many smart questions, and there are also some which are less so.
Your comment insinuates, but does not state, that this is wrong somehow. If that was your true intention then you are both smug and wrong.
Science is not something you should put in a glass case, only to be handled by ordained priests in the appropriate manner. Asking questions and providing answers to ones best abilities is how it is best interacted with.
If you know better, that is amazing. If you see wrong answers: correct them. If you see misleading questions: explain why they are so. Just don’t be smug, nothing ever got better by that.
People shouldn't be allowed to ask questions or make uneducated guesses to explore and understand the problem and satisfy their curiosity. We should ask you for permission first.
I'd even go so far to say that plenty people on these forums have the required knowledge in physics, mathematics, and computer science to make meaningful contributions to this conversation. This is not beyond the realm of every day people, it's a pretty accessible problem for many.
I guess this is downvoted because the initial conditions allude to numerical integration which isn't considered a proper "solution". Nevertheless, indeed the three-body (as well the n-body albeit with restrictions) has an analytic solution in form of power series. When people say it doesn't have one they mean in a closed-form. The research here isn't a solution (it doesn't allow one to find the exact state of the bodies at some specific later time) but rather a stochastic predictor of the behavior of the system.
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."