15 ms·

Over 150 new families of Newtonian periodic planar three-body orbits

89 points by msuvakov 9y ago

yorteiler 9y ago
What are the implications of this? Until 2013, only 3 or so solutions to the 3-body problem are known. Now we have over 150 solutions. This sounds incredible, given how fundamental the problem is, but So what? Will this change astrophysics - for example - in any way?
- ordu 9y ago
  3-body problem is unsolved math problem. There is equation, but no one knows how to solve it, and the only way to deal with it is number crunching. Stable orbits are solutions to the problem, but partial ones. The more solutions are known there more possibility to find more generalized solution. Maybe even the general solution for the problem. I do not know how it can change astrophysics (its just newtonian gravitation, not einshteinian), but it can bring new methods/ideas to mathematics, then improved math will change everything. Maybe.
  - reikonomusha 9y ago
    The problem isn't unsolved. It just doesn't have analytical solutions. But the mere fact we can simulate trajectories means we can solve it.
    - ordu 9y ago
      From engineering point of view, maybe the problem is solved. From maths point of view it isn't. For example, one cannot say for sure, are these trajectories periodic or no. 100-digit precision says "yes", but there is no guarantee, that 101-digit precision wouldn't say "no", they are not periodic.
  - 8note 9y ago
    what kinds of problems can we solve with an analytic solution to the 3 body problem instead of a numerical one?
- ooqr 9y ago
  Given this other reply: https://news.ycombinator.com/item?id=14280369 https://news.ycombinator.com/item?id=14280369 We'll probably be able to use these techniques on other problems in the future as well.
grw_ 9y ago
I, for one, welcome our new Trisolarian overlords!
- dvirsky 9y ago
  For anyone wondering - https://en.wikipedia.org/wiki/The_Three-Body_Problem https://en.wikipedia.org/wiki/The_Three-Body_Problem I'm just in the middle of the third book now, the series is well worth reading.
  - beached_whale 9y ago
    I cannot wait to see the movie, subtitles or dubs, the story was really interesting.
- ivan_gammel 9y ago
  Exactly what I was thinking about. Is the book a fiction or Chinese DO KNOW something?
MichailP 9y ago
I wonder how can authors claim such great certainty in their results, after all, it is all based on number crunching. Some floating point error, and similar is bound to creep in...
- privong 9y ago
  In the text they say they use high-precision floating point calculations to reduce this type of error: > At first, using the obtained initial conditions, we checked the 137 periodic orbits by means of the high-order Taylor series method in the 100-digit precision with truncation errors less than 10^−70 , and guaranteed that they are indeed periodic orbits. > Besides, we use the CNS with even smaller round-off error (in 120-digit precision) and truncation error (less than 10^−90 ) to guarantee the reliability of these 27 families. They do reference some software (e.g., "dop853"), but I'm not familiar with the details of those ODE solvers.
  - MichailP 9y ago
    That sounds really impressive. I jumped to comment before reading the paper. I found a nice gem in David Tongs lectures [1] in chapter about Dimensional analysis. He mentions Planck legnth Lp ~ 10^-35m, for which "All indications are that this is the shortest distance scale possible; at distances shorter than Lp, space itself is likely to have no meaning". So 10^-70 sounds like a nice margin. [1] http://www.damtp.cam.ac.uk/user/tong/relativity/dynrel.pdf http://www.damtp.cam.ac.uk/user/tong/relativity/dynrel.pdf
    - DamonHD 9y ago
      Lovely: a completely 'practical' bound on a totally theoretical numeric problem.
    - privong 9y ago
      > He mentions Planck legnth Lp ~ 10^-35m, for which "All indications are that this is the shortest distance scale possible; at distances shorter than Lp, space itself is likely to have no meaning". So 10^-70 sounds like a nice margin. That's an apples-to-oranges comparison, though. The 10^-70 number is a relative error while the planck length is just a length. The numerical computations in the paper were likely done in a system of "simulation units" where the actual lengths, velocities, forces, etc. are normalized to values of order unity. This has advantages for the numerical aspect, in terms of preserving floating point accuracy. But it also means that to translate it to a physical system (e.g., a triple system of stars), the simulation needs to be scaled to physical units. The 10^-70 that's quoted just means that the values should be numerically accurate to 1 part in 10^70. If you wanted to compare the (floating point) accuracy of the simulation with the Planck length, you would need to scale the simulation to a physical size and see what the 10^-70 fractional error would translate to in physical units.
JoeAltmaier 9y ago
Holy cow, some of those are complex. The equations of motion must be arcane.
- mathgenius 9y ago
  There very likely won't be any "equation of motion."
GregBuchholz 9y ago
Here's an older visualization for other interesting planar n-body choreographies: http://www.maths.manchester.ac.uk/~jm/Choreographies/ http://www.maths.manchester.ac.uk/~jm/Choreographies/ ...plus a link to the animations for the linked paper: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm http://numericaltank.sjtu.edu.cn/three-body/three-body.htm
- lanna 9y ago
  We live in a boring planetary system. Can you imagine if we lived in one of those choreographies how difficult would it be for Kepler to devise the laws of planetary motion?
  - pmiller2 9y ago
    The presence of a single body much more massive than the others (the sun) is what prevents that from happening.
  - kobeya 9y ago
    There is a popular (but IMHO terrible) Chinese science fiction book called "The three body problem" which tackles this exact issue.
    - Devagamster 9y ago
      I'd be interested to hear why you think that book is terrible
      - kobeya 9y ago
        For one, the science is dead wrong. Just because the three body problem doesn't have an analytical solution that doesn't mean that orbits are chaotic. Case in point: the very solar system in question is now known to have at least one planet in a very stable orbit. Nor does it mean that they are particularly difficult to calculate in many circumstances. You just can't use algebraic methods. So I was turned off because the what-if proposed was fantasy not science fiction. Fwiw I had the same response to the movie Arrival, which many people seemed to love. Second, the proposed solution that allows for the aliens to predict future seasons and therefore develop culture and technology reflects human problem solving. This is intellectually lazy and wrong. A more realistic solution, of the type I prefer to read about, would have the aliens evolve different ways of thinking that are more compatible with iterative methods than algebraic manipulation. To such a mind abstract reasoning would be really difficult, but it would be entirely natural to specify initial conditions and solve iteratively. Such a mind would find solving ordinary differential equations as easy as breathing, but solving x + 3 = 4 would need a computer. Even more interestingly, how would those differences in mind structure reflect the development of their language, culture, and societal structure? These are all very fascinating questions that seemed extremely obvious to me from the setup, and the sort of thing many hard sci-fi authors like Raynolds or Clarke would tackle. It's a book I would read and rave about. But instead we get the same old trope of basically human aliens invading earth because it is "habitable", never mind that habitability should be a non-concern to any self-respecting interstellar species, for whom climbing down a gravity well is probably a net negative. Planets are in fact shitty places to live in terms of galactic real estate, even more so when they are infected by foreign biology. Finally, the behavior of humans across the entire books is utterly unrelatable. I'm limited in how much I can say without spoiling the story. But in short every scientist is portrayed as incurious (wtf?) nihilist easily swayed by the suggestion to kill all humans. And the rest of humanity hardly dares better -- they willingly and dejectedly walk to their own execution when they are told to later in the series. I have no explanation of this other than bad writing. It was, in short, a really bad book.
GregBuchholz 9y ago
Has anyone explored the computational nature of Newtonian gravity? That is, if you carefully setup a set of masses in some manner, and let them interact though their gravitational pull, what kinds of things can you compute? Is gravity Turing complete? Is it a push down automata? Finite state machine? Can you use choreographies like these, coupled together to create register machines, or simulate cellular automata?
- hengheng 9y ago
  Not much of it should be robust to small input variations.
- kurlberg 9y ago
  Not quite a computation, but there is a striking example (due to Xia) where 5 particles interact and "kicks" off one particle to infinity in finite time. Further details can be found in http://www.ams.org/notices/199505/saari-2.pdf http://www.ams.org/notices/199505/saari-2.pdf
  - DamonHD 9y ago
    Interesting paper: I could just about follow it and the 'm3' particle accelerator is somewhat terrifying!
  - aisofteng 9y ago
    Of course, that is physically impossible.
- mathgenius 9y ago
  I think Terrence Tao did something like this for fluid mechanics.
  - IngoBlechschmid 9y ago
    Indeed, reported on at https://www.quantamagazine.org/20140224-a-fluid-new-path-in-grand-math-challenge/ https://www.quantamagazine.org/20140224-a-fluid-new-path-in-....
macawfish 9y ago
It'd be fascinating to see a harmonic analysis of these orbits. They seem (qualitatively) to exist on some edge of order and chaos.
isoprophlex 9y ago
Reminds me of Cixin Liu's 'Three Body Problem', about alien invaders that want to escape their own chaotic three-body planetary system by taking over ours... The first two books of the trilogy are just spectacular science fiction.
kmill 9y ago
This paper seems to be missing some related work, for instance Carles Simó. Greg Minton created a computer-assisted proof system for showing that there must exist a choreography with parameters within a certain distance of some given approximate parameters. This isn't just a matter of more floating point precision; it certifies that there is a critical point for action of the right kind. http://gminton.org/#gravity http://gminton.org/#gravity and http://gminton.org/#cap http://gminton.org/#cap Greg Minton also has a bunch of proved choreographies at http://gminton.org/#choreo http://gminton.org/#choreo
- theoh 9y ago
  Simó work was presented beautifully here, in Java: http://www.ams.org/samplings/feature-column/fcarc-orbits1 http://www.ams.org/samplings/feature-column/fcarc-orbits1
  - Asooka 9y ago
    Note: that only works in Internet Explorer (not Edge, of course), since all other browsers have disabled support for NPAPI plugins, including Java. I guess it shouldn't be too hard to implement a JVM in JavaScript that's good enough to run simple applets like this, but sadly nobody is doing it.
    - theoh 9y ago
      Yes, I guess you can, alternatively, still download a JRE with an "appletviewer" app.
      - Asooka 9y ago
        How can you run an embedded applet in an external viewer? Also, how is that more secure than just running it in the browser?
        theoh 9y ago
        https://en.wikipedia.org/wiki/AppletViewer https://en.wikipedia.org/wiki/AppletViewer "The applet viewer operates on HTML documents, but all it looks for is embedded applet tags; any other HTML code in the document is ignored. Each time the applet viewer encounters an applet tag in an HTML document, it launches a separate applet viewer window containing the respective applet." As far as security goes, I don't think appletviewer claims to be more secure that the browser. It has a similar security policy/sandbox.
Animats 9y ago
How stable are these orbits? Do they tend to degenerate into simpler forms over time? If they're stable, do we see any asteroid triplets in such configurations? Why not?
musgravepeter 9y ago
Guess I have some work to do to update my mobile app Three Body! (It presents galleries of solutions up to those found in 2013, and lets you explore your own initial placements) iOS: https://itunes.apple.com/us/app/threebody-lite/id951920756?mt=8 https://itunes.apple.com/us/app/threebody-lite/id951920756?m... Android: https://play.google.com/store/apps/details?id=com.nbodyphysics.threebodylite https://play.google.com/store/apps/details?id=com.nbodyphysi... The authors of the new solutions have published all the initial conditions - so hopefully it won't be too hard. (Although they use an 8th order RK integrator and the one I have is a regularizing Bulirsch-Stoer integrator).

89 points by msuvakov 9y ago

What are the implications of this? Until 2013, only 3 or so solutions to the 3-body problem are known. Now we have over 150 solutions. This sounds incredible, given how fundamental the problem is, but So what? Will this change astrophysics - for example - in any way?

ordu 9y ago
3-body problem is unsolved math problem. There is equation, but no one knows how to solve it, and the only way to deal with it is number crunching. Stable orbits are solutions to the problem, but partial ones. The more solutions are known there more possibility to find more generalized solution. Maybe even the general solution for the problem. I do not know how it can change astrophysics (its just newtonian gravitation, not einshteinian), but it can bring new methods/ideas to mathematics, then improved math will change everything. Maybe.
- reikonomusha 9y ago
  The problem isn't unsolved. It just doesn't have analytical solutions. But the mere fact we can simulate trajectories means we can solve it.
  - ordu 9y ago
    From engineering point of view, maybe the problem is solved. From maths point of view it isn't. For example, one cannot say for sure, are these trajectories periodic or no. 100-digit precision says "yes", but there is no guarantee, that 101-digit precision wouldn't say "no", they are not periodic.
- 8note 9y ago
  what kinds of problems can we solve with an analytic solution to the 3 body problem instead of a numerical one?
ooqr 9y ago
Given this other reply: https://news.ycombinator.com/item?id=14280369 https://news.ycombinator.com/item?id=14280369 We'll probably be able to use these techniques on other problems in the future as well.

grw_ 9y ago

I, for one, welcome our new Trisolarian overlords!

dvirsky 9y ago
For anyone wondering - https://en.wikipedia.org/wiki/The_Three-Body_Problem https://en.wikipedia.org/wiki/The_Three-Body_Problem I'm just in the middle of the third book now, the series is well worth reading.
- beached_whale 9y ago
  I cannot wait to see the movie, subtitles or dubs, the story was really interesting.
ivan_gammel 9y ago
Exactly what I was thinking about. Is the book a fiction or Chinese DO KNOW something?

MichailP 9y ago

I wonder how can authors claim such great certainty in their results, after all, it is all based on number crunching. Some floating point error, and similar is bound to creep in...

privong 9y ago
In the text they say they use high-precision floating point calculations to reduce this type of error: > At first, using the obtained initial conditions, we checked the 137 periodic orbits by means of the high-order Taylor series method in the 100-digit precision with truncation errors less than 10^−70 , and guaranteed that they are indeed periodic orbits. > Besides, we use the CNS with even smaller round-off error (in 120-digit precision) and truncation error (less than 10^−90 ) to guarantee the reliability of these 27 families. They do reference some software (e.g., "dop853"), but I'm not familiar with the details of those ODE solvers.
- MichailP 9y ago
  That sounds really impressive. I jumped to comment before reading the paper. I found a nice gem in David Tongs lectures [1] in chapter about Dimensional analysis. He mentions Planck legnth Lp ~ 10^-35m, for which "All indications are that this is the shortest distance scale possible; at distances shorter than Lp, space itself is likely to have no meaning". So 10^-70 sounds like a nice margin. [1] http://www.damtp.cam.ac.uk/user/tong/relativity/dynrel.pdf http://www.damtp.cam.ac.uk/user/tong/relativity/dynrel.pdf
  - DamonHD 9y ago
    Lovely: a completely 'practical' bound on a totally theoretical numeric problem.
  - privong 9y ago
    > He mentions Planck legnth Lp ~ 10^-35m, for which "All indications are that this is the shortest distance scale possible; at distances shorter than Lp, space itself is likely to have no meaning". So 10^-70 sounds like a nice margin. That's an apples-to-oranges comparison, though. The 10^-70 number is a relative error while the planck length is just a length. The numerical computations in the paper were likely done in a system of "simulation units" where the actual lengths, velocities, forces, etc. are normalized to values of order unity. This has advantages for the numerical aspect, in terms of preserving floating point accuracy. But it also means that to translate it to a physical system (e.g., a triple system of stars), the simulation needs to be scaled to physical units. The 10^-70 that's quoted just means that the values should be numerically accurate to 1 part in 10^70. If you wanted to compare the (floating point) accuracy of the simulation with the Planck length, you would need to scale the simulation to a physical size and see what the 10^-70 fractional error would translate to in physical units.

JoeAltmaier 9y ago

Holy cow, some of those are complex. The equations of motion must be arcane.

mathgenius 9y ago
There very likely won't be any "equation of motion."

GregBuchholz 9y ago

Here's an older visualization for other interesting planar n-body choreographies: http://www.maths.manchester.ac.uk/~jm/Choreographies/ http://www.maths.manchester.ac.uk/~jm/Choreographies/ ...plus a link to the animations for the linked paper: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm http://numericaltank.sjtu.edu.cn/three-body/three-body.htm

lanna 9y ago
We live in a boring planetary system. Can you imagine if we lived in one of those choreographies how difficult would it be for Kepler to devise the laws of planetary motion?
- pmiller2 9y ago
  The presence of a single body much more massive than the others (the sun) is what prevents that from happening.
- kobeya 9y ago
  There is a popular (but IMHO terrible) Chinese science fiction book called "The three body problem" which tackles this exact issue.
  - Devagamster 9y ago
    I'd be interested to hear why you think that book is terrible
    - kobeya 9y ago
      For one, the science is dead wrong. Just because the three body problem doesn't have an analytical solution that doesn't mean that orbits are chaotic. Case in point: the very solar system in question is now known to have at least one planet in a very stable orbit. Nor does it mean that they are particularly difficult to calculate in many circumstances. You just can't use algebraic methods. So I was turned off because the what-if proposed was fantasy not science fiction. Fwiw I had the same response to the movie Arrival, which many people seemed to love. Second, the proposed solution that allows for the aliens to predict future seasons and therefore develop culture and technology reflects human problem solving. This is intellectually lazy and wrong. A more realistic solution, of the type I prefer to read about, would have the aliens evolve different ways of thinking that are more compatible with iterative methods than algebraic manipulation. To such a mind abstract reasoning would be really difficult, but it would be entirely natural to specify initial conditions and solve iteratively. Such a mind would find solving ordinary differential equations as easy as breathing, but solving x + 3 = 4 would need a computer. Even more interestingly, how would those differences in mind structure reflect the development of their language, culture, and societal structure? These are all very fascinating questions that seemed extremely obvious to me from the setup, and the sort of thing many hard sci-fi authors like Raynolds or Clarke would tackle. It's a book I would read and rave about. But instead we get the same old trope of basically human aliens invading earth because it is "habitable", never mind that habitability should be a non-concern to any self-respecting interstellar species, for whom climbing down a gravity well is probably a net negative. Planets are in fact shitty places to live in terms of galactic real estate, even more so when they are infected by foreign biology. Finally, the behavior of humans across the entire books is utterly unrelatable. I'm limited in how much I can say without spoiling the story. But in short every scientist is portrayed as incurious (wtf?) nihilist easily swayed by the suggestion to kill all humans. And the rest of humanity hardly dares better -- they willingly and dejectedly walk to their own execution when they are told to later in the series. I have no explanation of this other than bad writing. It was, in short, a really bad book.

GregBuchholz 9y ago

Has anyone explored the computational nature of Newtonian gravity? That is, if you carefully setup a set of masses in some manner, and let them interact though their gravitational pull, what kinds of things can you compute? Is gravity Turing complete? Is it a push down automata? Finite state machine? Can you use choreographies like these, coupled together to create register machines, or simulate cellular automata?

hengheng 9y ago
Not much of it should be robust to small input variations.
kurlberg 9y ago
Not quite a computation, but there is a striking example (due to Xia) where 5 particles interact and "kicks" off one particle to infinity in finite time. Further details can be found in http://www.ams.org/notices/199505/saari-2.pdf http://www.ams.org/notices/199505/saari-2.pdf
- DamonHD 9y ago
  Interesting paper: I could just about follow it and the 'm3' particle accelerator is somewhat terrifying!
- aisofteng 9y ago
  Of course, that is physically impossible.
mathgenius 9y ago
I think Terrence Tao did something like this for fluid mechanics.
- IngoBlechschmid 9y ago
  Indeed, reported on at https://www.quantamagazine.org/20140224-a-fluid-new-path-in-grand-math-challenge/ https://www.quantamagazine.org/20140224-a-fluid-new-path-in-....

macawfish 9y ago

It'd be fascinating to see a harmonic analysis of these orbits. They seem (qualitatively) to exist on some edge of order and chaos.

isoprophlex 9y ago

Reminds me of Cixin Liu's 'Three Body Problem', about alien invaders that want to escape their own chaotic three-body planetary system by taking over ours... The first two books of the trilogy are just spectacular science fiction.

kmill 9y ago

This paper seems to be missing some related work, for instance Carles Simó. Greg Minton created a computer-assisted proof system for showing that there must exist a choreography with parameters within a certain distance of some given approximate parameters. This isn't just a matter of more floating point precision; it certifies that there is a critical point for action of the right kind. http://gminton.org/#gravity http://gminton.org/#gravity and http://gminton.org/#cap http://gminton.org/#cap Greg Minton also has a bunch of proved choreographies at http://gminton.org/#choreo http://gminton.org/#choreo

theoh 9y ago
Simó work was presented beautifully here, in Java: http://www.ams.org/samplings/feature-column/fcarc-orbits1 http://www.ams.org/samplings/feature-column/fcarc-orbits1
- Asooka 9y ago
  Note: that only works in Internet Explorer (not Edge, of course), since all other browsers have disabled support for NPAPI plugins, including Java. I guess it shouldn't be too hard to implement a JVM in JavaScript that's good enough to run simple applets like this, but sadly nobody is doing it.
  - theoh 9y ago
    Yes, I guess you can, alternatively, still download a JRE with an "appletviewer" app.
    - Asooka 9y ago
      How can you run an embedded applet in an external viewer? Also, how is that more secure than just running it in the browser?
      - theoh 9y ago
        https://en.wikipedia.org/wiki/AppletViewer https://en.wikipedia.org/wiki/AppletViewer "The applet viewer operates on HTML documents, but all it looks for is embedded applet tags; any other HTML code in the document is ignored. Each time the applet viewer encounters an applet tag in an HTML document, it launches a separate applet viewer window containing the respective applet." As far as security goes, I don't think appletviewer claims to be more secure that the browser. It has a similar security policy/sandbox.

Animats 9y ago

How stable are these orbits? Do they tend to degenerate into simpler forms over time? If they're stable, do we see any asteroid triplets in such configurations? Why not?

musgravepeter 9y ago

Guess I have some work to do to update my mobile app Three Body! (It presents galleries of solutions up to those found in 2013, and lets you explore your own initial placements) iOS: https://itunes.apple.com/us/app/threebody-lite/id951920756?mt=8 https://itunes.apple.com/us/app/threebody-lite/id951920756?m... Android: https://play.google.com/store/apps/details?id=com.nbodyphysics.threebodylite https://play.google.com/store/apps/details?id=com.nbodyphysi... The authors of the new solutions have published all the initial conditions - so hopefully it won't be too hard. (Although they use an 8th order RK integrator and the one I have is a regularizing Bulirsch-Stoer integrator).