
Over 150 new families of Newtonian periodic planar three-body orbits - msuvakov
https://arxiv.org/abs/1705.00527
======
GregBuchholz
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?

~~~
kurlberg
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)

~~~
aisofteng
Of course, that is physically impossible.

------
kmill
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
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
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
Yes, I guess you can, alternatively, still download a JRE with an
"appletviewer" app.

~~~
Asooka
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
[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.

------
GregBuchholz
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
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?

~~~
kobeya
There is a popular (but IMHO terrible) Chinese science fiction book called
"The three body problem" which tackles this exact issue.

~~~
Devagamster
I'd be interested to hear why you think that book is terrible

~~~
kobeya
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.

------
Animats
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?

------
isoprophlex
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.

------
musgravepeter
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?m...](https://itunes.apple.com/us/app/threebody-
lite/id951920756?mt=8) Android:
[https://play.google.com/store/apps/details?id=com.nbodyphysi...](https://play.google.com/store/apps/details?id=com.nbodyphysics.threebodylite)

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).

------
MichailP
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...

~~~
exDM69
The numerical methods used for this kind of calculations are engineered to
compensate numerical errors (floating point and errors inherent to
integrators).

They take advantage of a priori knowledge about the laws of physics, in
particular the conservation of mechanical energy and angular momentum.
Predictor-corrector is one family of methods. Other methods rely on
convergence as timesteps change.

There is a lot of literature on numerical integration applied to celestial
mechanics and new methods being released every year. These methods are
tailored to this problem space.

~~~
nonbel
When I looked into making a solar system simulation awhile back, I actually
found it shocking how much goes into monitoring and ensuring conservation of
energy/momentum in these models.

Once I was aware of this issue though, even more shocking to me was that many
papers on climate models I came across do not even mention they monitored
adherence to conservation laws. There is a disconnect there, I would think
this adherence either is a big deal (as would be suggested by the practice in
astronomy) or not (as would be suggested by climate research), not that it
would change by subfield.

~~~
strainer
When just doing point gravity dynamics (no friction, collision etc) energy and
momentum take care of themselves as long as the math is clean. I say this
confidently because I made a solar system simulation[1] with no attention to
conservation of energy at all. In practice Star systems also shed large
amounts of angular momentum through solar wind[2], but I expect this is often
not modelled.

Climate models are an entirely different matter and with respect, there is a
disconnect with your scepticism of them for this matter.

[1] [https://strainer.github.io/fancy#00](https://strainer.github.io/fancy#00)

[2]
[http://farside.ph.utexas.edu/teaching/plasma/Plasmahtml/node...](http://farside.ph.utexas.edu/teaching/plasma/Plasmahtml/node69.html)

~~~
nonbel
It says in your readme[1] though that your solar system simulation simulation
is off by 70k km (regarding Earth orbit) after only 1 year relative to the JPL
ephemerides, for "unknown reasons". I don't see how you can use that to
justify not checking for conservation of momentum/energy.

I also do not understand your final sentence.

[1] [https://github.com/strainer/fancy](https://github.com/strainer/fancy)

~~~
strainer
Come on it doesnt say "unknown reasons" it says the "It is possible the
discrepancy is due to the limitation of javascripts 64bit FP numbers rather
than subtle algorithmic or physics error"

70k km / a year (on a billion km long orbit) was a lot closer than I had hoped
for, considering its a 64 bit newtonian model. But this is beside the fact
that having worked on the project a fair bit, I understand that any model
adjustments to conserve energy or momentum are non-physical 'fudges' to
correct deficiencies of the model. There are no laws of physics dedicated to
adjusting energy, momentum, information in order to conserve them. They are
conserved by a mathematical correctness of every physical law - a proper
account of that eludes me, but it is certainly explained somewhere.

~~~
nonbel
Right, you don't know how much error the floating point error accounts for.
You also brought up loss of angular momentum to solar wind, etc. These are all
just speculations. That may be fine for the purposes of your simulation, but
not for predicting if asteroids will impact the earth or the influence of
human activity on Earth's climate.

Also, I know you are supposed to use a symplectic integrator to ensure
conservation of energy when doing these simulations, so it is not somehow
hardcoded into the laws, eg:
[https://en.wikipedia.org/wiki/Leapfrog_integration](https://en.wikipedia.org/wiki/Leapfrog_integration)

~~~
strainer
We could certainly calculate how much error was introduced by rounding, if we
had the time and inclination. It is useful to understand that about numerical
modelling, it should not be necessary to accept any indeterminable
discrepancies.

Conservation of energy, momentum, information _is_ hardcoded into all known
laws of Physics. It is our integration algorithms which do not all ensure
conservation of information - but that is carefully achievable with basic
first order schemes, such as Verlet and leapfrog integration.

~~~
nonbel
>"Conservation of energy, momentum, information is hardcoded into all known
laws of Physics. It is our integration algorithms which do not all ensure
conservation of information - but that is carefully achievable with basic
first order schemes, such as Verlet and leapfrog integration."

I am not sure about "conservation is encoded" part, but I can believe it.
Either way, in practice the simulations will encounter issues along these
lines if they are not careful. For example, I left your simulation running for
what amounted to ~120 yrs and saturn lost all her moons. This is the best pic
I could get, sorry (hopefully it is reproducible):
[https://i.imgur.com/C7emd28.png](https://i.imgur.com/C7emd28.png)

Since you do not report anything about conservation of energy/momentum, how do
we know that isn't the problem?

~~~
strainer
>For example, I left your simulation running for what amounted to ~120 yrs and
saturn lost all her moons.

Thanks! heh, it is rather complex there are all sorts of things... most likely
the new gravity function set for that model messed up, which can apply
attraction between groups of objects if they are distant and weak enough.

But honestly I am certain that it is something in particular and not something
mysterious about conservation of energy/momentum which i have not come across
yet, because there is no formula in that gravity simulation which _should_ be
able to destroy momentum. In the other models, there are quasi-physical
friction and pressure functions which do destroy momentum and do have some
rough adjustments to compensate for that somewhat, but accuracy is lost
whether things are compensated for or not. Tracking energy and momentum can be
a good way for tracking accuracy of a model, but if you do any adjustments to
_correct_ it, that is a kind of fudge not based in true physics. It is
poignant that you expressed scepticism of climate models because of a
perceived lack of such adjustments, while the opposite is actually true.
Climate models have to include a lot of calibrated adjustment and quasi-
physics, because the whole system is too complex to represent at all relevant
scales. This should not condemn the work of computer scientists specialised in
climate modelling to undue scepticism.

~~~
exDM69
> But honestly I am certain that it is something in particular and not
> something mysterious about conservation of energy/momentum which i have not
> come across yet, because there is no formula in that gravity simulation
> which should be able to destroy momentum

It's a well established fact that numerical integration methods either gain or
lose energy (in particular, the Runge-Kutta family is known to lose energy
over time). For celestial mechanics simulations, a special class of numerical
integration methods called "symplectic integrators" are used and their purpose
is to conserve energy and angular momentum.

> but if you do any adjustments to correct it, that is a kind of fudge not
> based in true physics.

When you are numerically integrating differential equations that model
physical phenomena, you're not doing "true physics" but an approximation
thereof.

And an approximation that makes Earth drift 70 km per year or Saturn's moons
drift out of orbit in a few hundred years is a very bad approximation by
scientific standards.

The methods used for celestial mechanics calculations need to be precise over
thousands to millions of years. And the way they work is to "fudge" with the
numerical methods to preserve energy and angular momentum. It's a much better
approximation of "true physics" than your toy simulation.

Your assumption that the issues are due to floating point errors is incorrect.
64 bit double precision is millimeter accurate to the orbit of Neptune. That's
good enough for scientific applications.

If you're interested in this, you could take a look at this scientific grade
N-body simulator and the methods it uses:
[https://github.com/hannorein/rebound](https://github.com/hannorein/rebound)

~~~
strainer
> Your assumption that the issues are due to floating point errors is
> incorrect.

Its not as easy as declaring the resolution of absolute position. If you
really want to spend the time on figuring it out, then examine the difference
in scale between the bodies positions, velocities, accelerations given
different model timestep values, what happens during the squaring summing and
rooting, multiplying by G of those values and also even (because i _have_
looked into this before) the subdivided timestep values involved in tempering
this models data, so it becomes a perfectly stable quantised version of its
anologe values (with no need for the kind of corrections Runge-Kutta is
notable for)

That link is very intresting to me thankyou. You can notice that it includes
some quite simple integration schemes as 'symplectic' (basically means stable
without need for gross correction)

------
yorteiler
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
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
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
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.

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

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

~~~
mathgenius
There very likely won't be any "equation of motion."

------
grw_
I, for one, welcome our new Trisolarian overlords!

~~~
dvirsky
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
I cannot wait to see the movie, subtitles or dubs, the story was really
interesting.

