
New periodic orbits of the three-body problem - msuvakov
https://phys.org/news/2017-10-scientists-periodic-orbits-famous-three-body.html
======
fdej
The "CNS" is as far as I can tell just the use of a high order Taylor series
together with multiple precision arithmetic for solving ODEs. This is a well
known method for long term simulation of dynamical systems that has been
around for decades. The authors basically imply that they invented this method
and were the first to be able to study long term evolution of dynamical
systems reliably because of it. Fair enough if they just weren't aware of
previous work (which happens all the time), but it's an oversight that
shouldn't have passed peer review.

That said, using this method to discover new periodic solutions of the three-
body system is a very neat application, which deserves applause!

~~~
spuz
Are you sure that their numerical method isn't new? The abstract says that the
author created it in 2009 and this paper further develops it to apply to the
three body problem. If it already existed, can you say why it had not already
been applied to the three body problem?

~~~
fdej
In fact the more general N-body problem is perhaps one of the most classical
uses of high order Taylor methods, in particular for studying the long term
stability of the solar system. Here is a paper from 1993:
[http://adsabs.harvard.edu/full/1993A%26A...272..687L](http://adsabs.harvard.edu/full/1993A%26A...272..687L).
Rigorous error analysis of Taylor methods for ODEs goes all the way back to
Moore's original work on interval arithmetic in the 1960s; in the early 2000s
Makino and Berz developed so-called Taylor models which combine rigorous error
bounds with accurate long-term propagation of changes due to perturbations in
the initial values (see
[http://bt.pa.msu.edu/index_TaylorModels.htm);](http://bt.pa.msu.edu/index_TaylorModels.htm\);)
they used it to study the dynamics of the solar system among other things. R.
Barrio and others have published several papers about reliable solution of
ODEs using Taylor methods; leading to the development of the TIDES software.
See several references listed on
[http://cody.unizar.es/tides.html](http://cody.unizar.es/tides.html). In
particular the paper
[https://doi.org/10.1016/j.amc.2004.02.015](https://doi.org/10.1016/j.amc.2004.02.015)
from 2005 investigates using variable order Taylor series with extended
precision for the solutions of dynamical systems. The paper
[http://dx.doi.org/10.1155/2012/716024](http://dx.doi.org/10.1155/2012/716024)
from 2012 is specifically about obtaining periodic solutions of dynamical
systems using a high order Taylor method with multiple precision arithmetic.

Actually, I made the first comment after I checked the second author's earlier
paper [https://arxiv.org/abs/1109.0130](https://arxiv.org/abs/1109.0130) where
they essentially made the claim about inventing "CNS" as the first-ever
reliable technique for long-term solutions of dynamical systems, without
referencing any of the earlier work I mentioned above.

However, I just checked the actual text of this new article on the three-body
problem (through sci-hub) and there the authors _do_ cite the earlier work I
mentioned above, giving proper attribution to others for the basic ideas
behind what they call "CNS"! So all is actually well, and the peer review
presumably did work (or the authors found out about the earlier work even
before writing the new paper). It is just the press release that is
misleading, as usual.

To answer your question, I'm not really familiar with the research on the
N-body problem, so I can't say why or whether no one tried looking for
periodic solutions in this way before. Perhaps no one actually thought of it,
or they didn't try since they didn't expect to find anything, or they didn't
have the computational resources, or they just couldn't figure out the details
of how to do it (which the present authors did, and deserve credit for).
Again, I was not trying to downplay the significance of this work, and finding
new applications of existing methods (and making even tiny improvements along
the way) is how science progresses. It also happens all the time that methods
get rediscovered/reinvented independently.

~~~
jcoffland
The novel part is not the use of Taylor series for simulation. CNS stands for
Clean Numerical Solution. The method of correcting the numerical error in the
simulation is what is claimed to be novel.

From the CSN paper:

"the residual and round-off errors are verified and estimated carefully by
means of different time-step Δt, different precision of data, and different
order M of Taylor expansion....for the considered problem, the truncation and
round-off errors of the CNS can be reduced even to the level of 10^−1244 and
10^−1000, respectively, so that the micro-level inherent physical uncertainty
of the initial condition (in the level of 10^−60) of the H\'{e}non-Heiles
system can be investigated accurately."

~~~
fdej
This is not a new idea. If you have a variable-order, variable-precision
implementation, it's the most obvious way to estimate the error of a numerical
solution.

------
ccleve
I'm halfway through The Three Body Problem, a science fiction novel by Chinese
author Cixin Liu. The apparent irrationality of three-body orbits is central
to the story. So far, it's excellent, both as science fiction and as social
commentary on contemporary China.

~~~
xpil
I've read it a couple of months ago. Interesting idea but terrible book
overall. Very naïve. Will definitely not read the second part.

~~~
mercer
Could you elaborate? I usually read positive stuff about this book, so I'm
very curious to hear diverging views.

------
forkandwait
> Today, chaotic dynamics are widely regarded as the third great scientific
> revolution in physics in 20th century, comparable to relativity and quantum
> mechanics.

Really??

~~~
twic
I wonder if that tells you something about the author's age. Chaos theory was
the next big thing in the '80s and '90s - recall that Jeff Goldblum played a
chaos theorist in Jurassic Park in 1993! It seems much less exciting today. I
think it gave way to string theory, and now we have AI (yet again).

~~~
woodandsteel
String theory is unproven, and AI is a technology, not a scientific theory.

------
est
The result videos from the authors is located at
[http://numericaltank.sjtu.edu.cn/three-body/three-
body.htm](http://numericaltank.sjtu.edu.cn/three-body/three-body.htm)

Funny thing is, it can not be opened, because of the 19th national congress.
All .edu.cn second-level domain were shutdown for "security reasons"

~~~
yorwba
Huh. This is the first time I can access a website from China, but not when
using a VPN. Here's a random GIF of orbital dynamics for you:
[https://imgur.com/kzPcL4h](https://imgur.com/kzPcL4h) (Mirror of
[http://numericaltank.sjtu.edu.cn/three-body/unequal-mass-
dat...](http://numericaltank.sjtu.edu.cn/three-body/unequal-mass-
data/I.A-0.5-4.gif))

------
sp332
The site hosting the videos is down. Are there only 6 families (as in the
caption at the top) or 600? Are all of the orbits discussed in the article in
2D or are some of them 3D? At the end it says the new CNS technique found only
243 new families, so how were the hundreds of other new ones found?

~~~
sova
CNS from the article, " clean numerical simulation" enabled them to find 243
more than what the computational power was capable of finding with a lossy
representation of periodicity. In other words, the strength of the machine
using old-style algorithms for detection would have gotten 357+ solutions, and
their use of CNS which cleans up the math a lot for the supercomputer helped
them find an additional 243 that would have slipped through the cracks.

------
musgravepeter
The pre-print from arxiv:
[https://arxiv.org/abs/1705.00527v4](https://arxiv.org/abs/1705.00527v4)

------
phkahler
Are any of them dynamically stable?

~~~
Pxtl
Yeah, that's what I'm curious about. Would all of them deviate if they were
breathed on wrong? Could any of them occur in nature?

~~~
phkahler
There are hints of that concern in TFA where they state the importance of
accurate simulations over a long time. On the other hand, a simple two body
system is very stable but will diverge in a simulation using a simple Euler
integrator.

~~~
Lerc
So here's an odd thing. I made a puzzle game for Android where I use Euler,
and some of the puzzles require the divergance to work. It has caused a
curious phenomenon where some people think the game is wrong, yet the game at
no stage mentions what the rules of motion are.

I have seen people talk about the puzzles in terms of planets, gravity,
massless bodies etc. when in-fact they are just circles that move towards or
away from other circles. Theoretically people who can identifiy the Euler
intergrator should be able to go "aha! I know what to do now" but so many get
hung up on the idea that it is wrong.

~~~
mnw21cam
Are you allowed to tell us the name of the game, so we can have a look?

------
davesque
From a complete layman's point of view, I wonder if it's appropriate to think
of a three-body system as somehow irreducible. In other words, maybe a
"closed-form" representation of three-body motion can be defined in terms of a
finite combination of stable periodic configurations. Anyway, just some
musings.

~~~
sova
Irreducible is a good way to describe the equally-balanced constraint in a
3-body situation.

------
powertower
Is there an upper limit on how many different periodic orbitals a 3-body
system may have?

~~~
ChuckMcM
Generally if you could prove that there was, or that their wasn't, you would
probably win the Fields Medal at least.

------
sizzzzlerz
So if the problem was first proposed by Newton in the 17th century, how did
early investigators simulate the interactions between the bodies up until the
electronic computer became available. There are no closed form solutions
A.F.A.I.K. so they must have calculated the body's state by hand, I guess. How
tedious.

~~~
rwmj
My probably naive intuition says that if you had a system with two "suns" in
the centre rotating around each other very closely, and one distant planet
rotating about the centre of mass of the two suns, that would be stable
surely?

~~~
musgravepeter
Yes, that does work. In the literature this is called a "heirarchical system"

The very simplest version is called the Euler problem. Two fixed masses and a
third moving in the "dipole field". All the solutions can be explicitly
determined (although only in terms of e.g. Jacobi elliptic functions and other
elliptics). There's a book "Integrable Systems in Celestial Mechanics" by
Mathuna.

I recently added the Euler problem to my iOS app ThreeBody:
[https://appadvice.com/app/threebody-
lite/951920756](https://appadvice.com/app/threebody-lite/951920756)

Some day I'll get around to adding these 600 solutions...

~~~
adrianratnapala
Right, but this dipole solution will neglect the perturbation of the third
body on whatever is creating the dipole. Fair enough, because we expect the
effect to be tiny.

But still you would have to unpack that dipole approximation to figure out if
this perturbation will slowly change it in ways that do something significant
in the long run.

------
da-bacon
You can find a bunch of n-body solutions, some of which like those in this
article are pretty amazing, on Cris Moore's gallery page
[http://tuvalu.santafe.edu/~moore/gallery.html](http://tuvalu.santafe.edu/~moore/gallery.html)

------
trhway
as we know it is too late, trisolaris fleet is already under way, and the
problem with the human science progress is so apparent that it even became a
subject of the recent Big Bang Theory episode.

~~~
lnanek2
fortunately, we live in a dark forest, and need only threaten to shine a light
to make the invasion fleet pointless...

~~~
mirimir
Yeah, but that doesn't always turn out so well :(

