Newton’s three-body problem explained - Fabio Pacucci

5,979,544 views ・ 2020-08-03

TED-Ed


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Reviewer: Nika Kotnik
00:07
In 2009, two researchers ran a simple experiment.
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Leta 2009 sta dva raziskovalca izvedla preprost poskus.
00:11
They took everything we know about our solar system
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Upoštevala sta vse, kar vemo o našem Sončnem sistemu,
00:15
and calculated where every planet would be up to 5 billion years in the future.
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in izračunala, kje se bo vsak planet nahajal do pet milijard let v prihodnost.
00:21
To do so they ran over 2,000 numerical simulations
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Za to sta izvedla preko 2000 numeričnih simulacij,
00:25
with the same exact initial conditions except for one difference:
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pod popolnoma enakimi začetnimi pogoji, z eno samo razliko:
00:29
the distance between Mercury and the Sun, modified by less than a millimeter
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razdaljo med Merkurjem in Soncem sta za vsako simulacijo
00:35
from one simulation to the next.
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spremenila manj kot za milimeter.
00:37
Shockingly, in about 1 percent of their simulations,
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Šokantno: pri približno odstotku teh simulacij
00:41
Mercury’s orbit changed so drastically that it could plunge into the Sun
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se je Merkurjeva orbita spremenila tako zelo, da bi strmoglavil v Sonce
00:46
or collide with Venus.
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ali trčil v Venero.
00:48
Worse yet,
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Še huje,
00:49
in one simulation it destabilized the entire inner solar system.
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v eni simulaciji je destabiliziral celoten notranji del Osončja.
00:54
This was no error; the astonishing variety in results
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To ni bila napaka; izjemna razlika v rezultatih
00:58
reveals the truth that our solar system may be much less stable than it seems.
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razkriva, da je naš Sončni sistem morda veliko manj stabilen, kot se zdi.
01:05
Astrophysicists refer to this astonishing property of gravitational systems
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Astrofiziki tej neverjetni lastnosti gravitacijskih sistemov
01:10
as the n-body problem.
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pravijo problem n-teles.
01:12
While we have equations that can completely predict
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Medtem ko enačbe lahko popolnoma točno napovejo
01:15
the motions of two gravitating masses,
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gibanje dveh teles pod vplivom težnosti,
01:17
our analytical tools fall short when faced with more populated systems.
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naša analitična orodja pri sistemih z več telesi ne delujejo.
01:23
It’s actually impossible to write down all the terms of a general formula
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Pravzaprav je nemogoče zapisati vse pogoje za splošno formulo,
01:28
that can exactly describe the motion of three or more gravitating objects.
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ki bi opisala gibanje treh ali več teles pod vplivom težnosti.
01:34
Why? The issue lies in how many unknown variables an n-body system contains.
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Zakaj? Težava je v številu neznank v sistemu n-teles.
01:41
Thanks to Isaac Newton, we can write a set of equations
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Zaradi Isaaca Newtona znamo zapisati set enačb,
01:45
to describe the gravitational force acting between bodies.
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ki opišejo silo težnosti med telesi.
01:49
However, when trying to find a general solution for the unknown variables
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Vendar se pri iskanju splošne rešitve za neznanke
01:53
in these equations,
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v teh enačbah
01:55
we’re faced with a mathematical constraint:
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srečamo z matematično omejitvijo:
za vsako neznanko mora obstajati vsaj ena enačba,
01:58
for each unknown, there must be at least one equation
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02:01
that independently describes it.
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ki jo neodvisno opiše.
02:04
Initially, a two-body system appears to have more unknown variables
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Na začetku se zdi, da ima sistem dveh teles več neznank
02:08
for position and velocity than equations of motion.
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za položaj in hitrost, kot ima enačb za gibanje.
02:12
However, there’s a trick:
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Toda trik je v tem:
02:14
consider the relative position and velocity of the two bodies
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upoštevajte relativni položaj in hitrost teh dveh teles
02:18
with respect to the center of gravity of the system.
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glede na težišče sistema.
02:22
This reduces the number of unknowns and leaves us with a solvable system.
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To zmanjša število neznank in omogoči rešitev sistema.
02:27
With three or more orbiting objects in the picture, everything gets messier.
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Težava pa nastane s tremi ali več telesi, ki medsebojno krožijo.
02:33
Even with the same mathematical trick of considering relative motions,
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Tudi z istim matematičnim trikom uporabe relativnih gibanj
02:37
we’re left with more unknowns than equations describing them.
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imamo več neznank kot enačb, ki jih opisujejo.
02:42
There are simply too many variables for this system of equations
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Za tak sistem enačb je enostavno preveč spremenljivk,
02:46
to be untangled into a general solution.
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da bi jih razvozlali v splošno rešitev.
02:49
But what does it actually look like for objects in our universe
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Kaj pa pravzaprav pomeni, če se telesa v vesolju
02:53
to move according to analytically unsolvable equations of motion?
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premikajo po analitično nerešljivih enačbah gibanja?
02:58
A system of three stars— like Alpha Centauri—
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V sistemu treh zvezd, denimo Alfa Centauri,
03:01
could come crashing into one another or, more likely,
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bi te lahko trčile ena v drugo, ali, kar je bolj verjetno,
03:05
some might get flung out of orbit after a long time of apparent stability.
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katera od njih bi morda odletela iz orbite po dolgem času stabilnosti.
03:10
Other than a few highly improbable stable configurations,
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Razen nekaj skrajno neverjetnih stabilnih konfiguracij
03:14
almost every possible case is unpredictable on long timescales.
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je skoraj vsak mogoč primer na dolgi rok nepredvidljiv.
03:20
Each has an astronomically large range of potential outcomes,
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Vsak ima astronomsko velik razpon možnih izidov,
03:24
dependent on the tiniest of differences in position and velocity.
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odvisno od najmanjših sprememb položaja in hitrosti.
03:29
This behaviour is known as chaotic by physicists,
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Takemu obnašanju fiziki pravijo kaotično
03:33
and is an important characteristic of n-body systems.
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in je pomembna značilnost sistema n teles.
03:37
Such a system is still deterministic— meaning there’s nothing random about it.
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Tak sistem je še vedno determinističen, se pravi, da na njem ni nič naključnega.
03:42
If multiple systems start from the exact same conditions,
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Če več sistemov začne pod povsem enakimi pogoji,
03:45
they’ll always reach the same result.
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bodo vedno prišli do istega rezultata.
03:48
But give one a little shove at the start, and all bets are off.
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Ampak če enega na začetku le malo sunemo, se lahko zgodi karkoli.
03:53
That’s clearly relevant for human space missions,
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To je seveda pomembno za odprave v vesolje,
03:57
when complicated orbits need to be calculated with great precision.
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ko je treba zapletene orbite zelo natančno izračunati.
04:02
Thankfully, continuous advancements in computer simulations
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Na srečo stalni razvoj računalniških simulacij
04:06
offer a number of ways to avoid catastrophe.
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na več načinov pomaga odvrniti katastrofe.
04:09
By approximating the solutions with increasingly powerful processors,
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S približnimi rešitvami s pomočjo vse močnejših procesorjev
04:13
we can more confidently predict the motion of n-body systems on long time-scales.
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lahko z večjo gotovostjo napovemo gibanje sistemov n teles na dolgi rok.
04:19
And if one body in a group of three is so light
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In če je eno telo v sistemu treh tako lahko,
04:22
it exerts no significant force on the other two,
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da na ostali dve ne izvaja pomembne sile,
04:25
the system behaves, with very good approximation, as a two-body system.
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se sistem obnaša zelo približno tako kot sistem dveh teles.
04:30
This approach is known as the “restricted three-body problem.”
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Ta pristop je znan kot “omejen problem treh teles”.
04:34
It proves extremely useful in describing, for example,
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Zelo uporaben je denimo pri opisovanju
04:38
an asteroid in the Earth-Sun gravitational field,
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asteroida v težnostnem polju Zemlje in Sonca
04:41
or a small planet in the field of a black hole and a star.
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ali majhnega planeta v polju črne luknje in zvezde.
04:46
As for our solar system, you’ll be happy to hear
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Za naš Sončni sistem vam z veseljem povem,
04:49
that we can have reasonable confidence in its stability
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da smo dokaj prepričani v njegovo stabilnost,
04:52
for at least the next several hundred million years.
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vsaj za naslednjih nekaj sto milijonov let.
04:56
Though if another star,
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Če pa je na poti nova zvezda,
ki prihaja z druge strani galakcije,
04:58
launched from across the galaxy, is on its way to us,
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se lahko zgodi karkoli.
05:02
all bets are off.
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