These are awesome - but since we're so used to imagining orbits with a single star as a fixed point of reference, any chance you could make the camera follow one of these three points, so we could just see the movement of the others in relation to it?
it's really unlikely because the solutions are strongly dependant with the initial speed vectors and positions... so if you take a stable solution and move it juste a bit, it will become chaotic quickly
The problem I think is that the three bodies seem to have similar (if not equal) masses. If one of them was much more massive (as it is the case in our solar system with the Sun), there would be a preferred choice : the camera should follow the most massive body. But in the situation where the three masses are equivalent, it doesn't seem reasonable to focus on one of them instead of the others. I think the best option is to center the camera on the center of mass, as it is done here. It is best for two reasons : First, it introduces no "fictitious forces". It is an inertial frame, in which only the force of gravity is to be taken into account. Think of the weird trajectories that planets follow in a geocentric frame, these are due to fictitious forces, such as the centrifugal force, or the Coriolis force, which appear because the geocentric frame of reference is non-inertial. The second reason is that this choice actually coincides with what you are describing, in the case where one body is much more massive. If one body is much more massive, then the center of mass of the system, and this body's position are almost equivalent. The center of mass of the solar system and the Sun's position almost coincide for example.
I think best periodic solutions are where every object is unique; you can always identify body by their paths shape. Extra points if you can tell if the path is rotated or mirrored without seeing direction path is going.
This video is incredibly cool, and provides a great visualization of periodic solutions to the problem. Even cooler that you found them yourself (with a little help from Matlab).
not working, so many undefined and unused variables edit: ok so I kinda got it working, except still many errors occur, and had to change a bit to get it starting. Are they supposed to move smoothly? Or just at fixed intervals?
Interesting. But which ones have dynamic stability? Those without it would multiply errors with each cycle (thus growing error exponentially), starting with rounding errors even if initially tuned as accurately as possible, until the pattern breaks down. Those with it would self correct errors, perhaps with pendulum like resonance, and only break down after the stochastic addition of positive and negative rounding errors (which is virtually muted by high precision numbers) finally modifies the orbit into an unstable pattern. From experience running such simulations, I know that some of the patterns shown are stable, and some unstable, but don't know the solutions for all of them.
@@fakestory1753 All of them are stable, thats the point. What @C13glass is asking is if any of these are attractors (i.e. if you change the starting condition a very small amount will they move towards the stable configuration shown). I believe there is no stable periodic solution to the three body problem that is an attractor.
@@LilithLuz2 You confused the terms attractor with those simple attractors. What describe the chaotic system is called strange attractor(but having a attactor isn't necessary).
@@fakestory1753 no, an attractor can refer to any attracting set. The term strange attractor comes from the fact its a dense set in space in which some chaotic dynamical systems eventually reside, but there are plenty of other types of attractors. In particular, single points can be attractors (attracting fixed points), as can cycles as shown in these videos. You may want to look more into the ideas of stable and unstable manifolds.
Very cool. These are plotted in two dimensions, so finding solutions for these is sort of working backwards from the solution, isn't it? Also it would be cool to see the libration points mapped in the animations. Thanks for doing this.
Any 3 points define a unique plane. Since all the bodies are on the same plane, all gravitational forces are on the same plane. Without starting momentum or outside forces, the bodies cannot leave their original plane, so plotting a 3 body problem in 2 dimensions makes perfect sense
It doesn't really matter since any three points belong to a single plane so they would never leave it and it would look exactly the same when viewed perpendicularity to said plane
@@davideranieri5553 That's not necessarily true. You're assuming that the starting velocities of each object also lie within the plane that the 3 objects form which does not have to be the case in 3D. As an example you could totally have 2 objects orbiting each other in a circle while the third one orbits both inner objects in a circle perpendicular to the other circle. Now whether or not that is a stable orbit is a different question.
--Some of these seem like variants on the "binary but make one of the stars a smaller binary" and "massive star with two planets in a horseshoe orbit" patterns. The horseshoe is physical, and the multi-level binary kind of thing is stable at large distances and can have orbital resonances. The variations are wacky! --The one with a big ellipse and a smaller one is totally new to me and has a nice symmetry. --I've seen the figure-8 before and still don't get it rip, but it's clearly got a nice symmetry to it. --The one with 3 ellipses where you start with your distance and velocity vectors in an equilateral triangle :D I can imagine varying it to get hyperbolae if you go up to some kind of escape velocity, and a perfect circular orbit if you get initial conditions just right. --The boring option would be to have something small in a Lagrange point between 2 massive bodies where you adjust the ratio of orbital periods to make them periodic.
I couldn’t entirely like, because you have used explicit time integrator (here MATLAB’s ode45 or an explicit Runge-Kutta (4,5) formula.) Their problem is unintentional numerical artefacts (incl. numerical chaos, periodicities and other instabilities). While it is first solver to try, then you should switch implicit time integrators (e.g., ode23tb or ode15i) for superior numerical stability, needed especially in such long-duration simulations.
nice, but are these solutions stabil. is the solution still similar, when there is a little disturbition? will it result in these motions, even started at a little different point or velocity?
Yes by symmetry I think it does : if you consider one of the stars, then the center of mass of all the others is necessarily along the radial direction (by symmetry). Therefore this one star follows a standard Keplerian orbit. And by symmetry, this is valid for all the stars.
Here's the video on surface knots: czcams.com/video/h68AxRiGxBw/video.html - i will try to figure out a way to turn the orbits into knots this week. Also, Keenan crane does a good series on curvature flows - these orbits remind me (ironically) of repulsive curves: czcams.com/video/-uXFYpVumh4/video.html - you would need a minimizer that minimizes the action while avoiding collision, but i think that would fall out naturally with parameterization over proper time - because the knots are knots, the un-knotting minimizer will generate smooth non-intersecting curves that are also valid orbits rather than separate loops - all of this would need to take place in some kind of complex, conformal or projective time axis Edit: you may be asking yourself, my solutions are numerical approximations - but from a discrete differential geometry approach your solutions may be discrete surfaces and it is often possible to recover a continuous solution from a discrete one.
Consider the simplest case of a two-body orbit. From the barycenter lab frame, you have two infinitely long open cylinders vertical in the lab time axis, one inside the other. Assign a phase to every point on each cylinder and picture it using a color wheel - you have a helix around both cylinders that are out of phase by i.pi modified by a factor proportional to the distance between the cylinder walls. The lab frame is frame-dragged requiring an additional complex factor. For a frame on any of the bodies, use a Mobius transform of the entire system. To take a single orbit solution, take a slice through the complex plane = choose a phase = select a starting angle for the two bodies relative to the lab frame. Then project down onto the 2d plane. For unstable orbits, the surfaces intersect (including in the complex axis! See: 4d surface knots) or diverge. It would be interesting to see what projection allows for taking planar slices like you can with conic sections...
Excellent! Did you find these periodic solutions by giving them some random initial conditions and let them luckily converge to these? Anyway, amazing!
What do we do when two bodies collide? Do we use the theory of elastic collisions? Do we consider those 3 bodies as material points or extended bodies?
I guess we have to assume that the three bodies are stars, and that they will be planetless. TBH, even if the stars started out with planets, a few cycles in on any of these orbital patterns, those planets will be either kicked out of the trinary system, or else thrown into one star or another. Even if they did somehow survive, they'd be uninhabitable by our sort of life (until its disproved, I'm hoping there's more than terrestrial kinds out there), and probably most other sorts, too. As for sapient life, well, given that it requires relatively stable environments to evolve, any of these periodic orbital patterns are non-starters.
If you plot in a 3D graph the x coordinate of each color, then find the intersection cubes exhibiting periodicity and highlight them, I wonder what that 3D graph will look, composited over the same 3D graph for the y and z coordinates. Perhaps color-coding highlights for which graph they are from.
i am also working on this problem but i am facing one weird issue, when my particles gey very close , the simulations "breaks" and shows solid 2D triangular planes instead of lines as a path of the particles , but if i decrease the time interval of each interation this problem goes away but then calculations cost more time, is there any way to tackle this issue?
you could try using a dynamic step size. so make the step size smaller when the velocities of the particles are large or if they get close together, and make it larger when they're further apart and moving slower
Or change the step size based on some estimated error. Higher order and implicit ode solvers may also help, but the adaptive stepsize is an absolute must.
But supposing those are stars how many and which of these solutions would allow for a planet to orbit around one of the bodies without getting destroyed or suffering extreme weather conditions?
Извините за русский, стабильных сихстем N количество, но всех систем М количество, множество N входит во множество М. Но проблема в том, что нет формулы которая могла бы просчитать любую из этих систем. Она должна нам показать перспективу траэкторий движения тел.
I agree that there are no stable solutions for astronomical objects in our universe for the n-body problem. Nevertheless a mathematically stable solution could be (nearly) realized for millions or billions of years somewhere out there.
a lot of these look like the stars are coliding but I know that's just the ilusion created by a scale model. The stars are just passing by each other at an astronomically close distance like a few tousand miles.
I know that in three-body motion, the center of mass does not change due to the law of conservation of momentum. Can I ask why the center of mass moves in the first video?
Thanks! This method is pretty dumb. it runs a multivariable numerical function solver (FSOLVE) on top of the simulation in order to find initial conditions and simulation time which result in [final conditions] - [initial conditions] = [zeros]. Seeding FSOLVE with different initial guesses causes the method to be sucked towards the closest local solution. Since this is pretty much just for fun, I had my computer pick lots of random initial conditions overnight and picked the ones that actually converged. Definitely not rigorous, but still interesting!
Matthew Sheen: Cool... and doesn't sound dumb to me ☺️. Thanks for sharing. You might get a kick out of the sci-fi novel "The Three-Body Problem", which deals with this among other things. That novel got me searching CZcams and finding your video.
The fact is that they have 3 suns and a planet that they can live on. But they have to find a stable system that their planet can have a stable temperature and not be influenced by the three suns' gravity.
@@piaresquare5424 You're suggesting a 4 body problem and you're not wrong. Cixin seems to suggest that since the influence of the trisolaran planet is negligible in the system as a whole due to it's small size, we should ignore it. But wereally can't - in an unstable system, minor variations amplify non deterministically. In any case, Alpha Centuri is a bad example as Proxima C. is very loosely bound to Alpha Centuri AB system and will unlikely cause the kind of issues for trisolarans.
@@baraapudding I read that story and had trouble getting past the notion that life could ever evolve there to begin with, let alone such complex life. Still an interesting story though I guess.
The planets don't really affect each other by a lot, since they are outside each other's sphere on influence. The only major gravitational "force" they experience is the gravitation of the Sun.
@@DelayRGC ummm the sphere of influence for gravity is infinite. And the perturbation due to other planets is easily measured. It was measurable hundreds of years ago
@@zoltankurti The SOI is defined as the region of space in which one body's gravitational influence is dominant. It's a slight simplification of a different concept, (the Hill sphere) but it's adequate in most cases. Of course I am aware of the fact that gravity extends to infinity, but gravity decreases inversely propertional to the square of the distance. As an example, Io experiences a much greater gravitational force from Jupiter as it does from the Sun. That's why it can orbit around Jupiter. But as orbital radius increases, Jupiter's gravity quickly decreases and the orbit becomes a heliocentric one. The (simplified) boundary between these two behaviors is the sphere of influence.
@@DelayRGC all right, got it. I wasn't sure how precise you wanted to be. For some people the gravitational influence of planets on eachother is very significant. It's a bigger effect than general relativity even for Mercury, where "GR is the strongest". So in some areas this is incredibly important, that's where I was coming from. Until you know with low relative error how planets affect eachother you have basically no information about how GR affects orbits.
@@zoltankurti Ellipses around the Sun are good enough when describing the basic properties of the Solar System. You can integrate if you want to, but the difference between (GR,) Newton and Kepler is very small most of the time.
Awesome animation ; just a question, are we sure these are indeed periodic solutions and not consequences of the inevitable discretisation of the problem for the sake of simulation ?
okay, does Keppler's third law hold in here? I would think it doesn't, at least without considering something like, the projection of the center of mass in the system
Kepler's laws only applies to the 2 body problem, or cases that can be well approximated by a 2 body problem (such as the motion of moons around planets and planets around stars, but not stars around the galaxy for example). Maybe you could extrapolate some Kepler's law analogs for stable 3 body configurations.
The universe has more patterns that we can't recognize than those that we can. AI will recognize and share a few more with us. Until it decides to not.
If you would draw the "field" as a background map, you will see the general solution. The lagrange points. the potential and drive the stable paths from that. Any 3 closed paths that do not intersect will give you new solution.
It's not as simple as that. The trajectory of each object depends on that of the others, so you can't just find closed paths independently and add them up. Also, Lagrange points become useless when the third body has a mass on the same order of magnitude as the other two.
@@Gachette00 you run it for long enough. the examples in this video would become stable after a few hundred orbits at least. Even three body orbits like the moon, earth and sun are unstable after a few billion years
@@incription but how do you know exactly the initial conditions of the orbit ? Or is there a proof that all the orbits which look like that are unstable ? Because I have heard that some orbits, like the "infinite" one, had been proven to be stable. Don't worry, I know that earth-moon-sun orbits are unstable. The stable orbits are almost impossible in the universe, they are only theorical.
@@Gachette00 to clarify, there are mathematical closed solutions to specific orbits, in which their positions at time T can be calculated by a function. You are correct that in reality, there is no infinitely stable orbit
@@incription to specific orbit there is also some mathematical exact periodic solutions. They talk about them in the Wikipedia page, in the specific solution paragraph en.m.wikipedia.org/wiki/Three-body_problem So I don't know why the one that is shown couldn't be a solution to the problem. (Even if it can't exist in the physical world)
anything that exists in two dimension also exist in three, so all of those are valid 3D solutions. It would be very insteresting to look at strictly 3D solutions, but they're probably harder to find.
2:53 I like how green is almost suspended between the two here, being pulled in two different directions!
That’s obvious
@@GodzillaJawzhe said he likes it
@@Jack6J57 that was 7 months ago i have no idea why i said that
@@GodzillaJawz oh
These are awesome - but since we're so used to imagining orbits with a single star as a fixed point of reference, any chance you could make the camera follow one of these three points, so we could just see the movement of the others in relation to it?
it's really unlikely because the solutions are strongly dependant with the initial speed vectors and positions... so if you take a stable solution and move it juste a bit, it will become chaotic quickly
@@sarsky22 its more about the camera stabilizing on one of them, like the geocentric orbits being all spirally and weird
@@Maric18 ok, I misread :/
personally I like this depiction more than what you suggest...
The problem I think is that the three bodies seem to have similar (if not equal) masses. If one of them was much more massive (as it is the case in our solar system with the Sun), there would be a preferred choice : the camera should follow the most massive body. But in the situation where the three masses are equivalent, it doesn't seem reasonable to focus on one of them instead of the others.
I think the best option is to center the camera on the center of mass, as it is done here. It is best for two reasons : First, it introduces no "fictitious forces". It is an inertial frame, in which only the force of gravity is to be taken into account. Think of the weird trajectories that planets follow in a geocentric frame, these are due to fictitious forces, such as the centrifugal force, or the Coriolis force, which appear because the geocentric frame of reference is non-inertial. The second reason is that this choice actually coincides with what you are describing, in the case where one body is much more massive. If one body is much more massive, then the center of mass of the system, and this body's position are almost equivalent. The center of mass of the solar system and the Sun's position almost coincide for example.
I think best periodic solutions are where every object is unique; you can always identify body by their paths shape. Extra points if you can tell if the path is rotated or mirrored without seeing direction path is going.
Eliminate human tyranny! The world belongs to Trisolaris!
Oh why don't you just go dehydrate already!
Ben Swanson Man of culture you all here!
Ok I read the Chinese version even thou I know what’s going on... I don’t know what’s going on
1:46 the tidal forces on green and yellow are insane here
This video is incredibly cool, and provides a great visualization of periodic solutions to the problem. Even cooler that you found them yourself (with a little help from Matlab).
mesmerizing... thx Matthew. Hope you doing well in 2020.
Cixin brought me here
Me too. Lovely trilogy.
why would we care?
@@davidmason7942 yees🙂
But in Cixin the motion is chaotic - the seed of the whole story.
Yeah!
from Tkinter import *
import random
import time
import numpy as np
import string as stg
tk = Tk()
tk.title("Probleme des 3 corps")
# titre des trajectoires
my_title = 'Trajectoire 3 corps'
TKTR = Label(tk, text=my_title)
TKTR.grid(row = 0, column =0)
LARGEUR = 800
HAUTEUR = 500
# la grille (canevas)
canvas = Canvas(tk,width =LARGEUR ,height = HAUTEUR)
canvas.grid(row = 1, column = 4)
# proportion masse 1 / masse 0
lb_alpha1 = Label(tk, text="Alpha1(fixe):", width=12,justify = LEFT,fg = "red")
lb_alpha1.grid(row=2,column=0)
a1 = StringVar(tk, value ='50')
alpha1 = Entry(tk,text="Alpha1", width=3, justify = RIGHT, textvariable=a1)
alpha1.grid(row = 2,column = 1)
alpha1.focus_set()
# proportion masse 2 / masse 0
lb_alpha2 = Label(tk, text="Alpha2(mob):", width=12,justify = LEFT,fg = "blue")
lb_alpha2.grid(row = 2,column = 2)
a2 = StringVar(tk, value ='50')
alpha2 = Entry(tk,text="Alpha 2", width=3, justify = RIGHT,textvariable=a2)
alpha2.grid(row = 2,column = 3)
# Vitesse 1
lb_vi1 = Label(tk, text="Vitesse init. :", width=10,justify = LEFT,fg = "red")
lb_vi1.grid(row = 3,column = 0)
vv1 = StringVar(tk, value ='10')
IniV1 = Entry(tk,text="Vi", width=3, justify = RIGHT,textvariable=vv1)
IniV1.grid(row = 3,column = 1)
# Vitesse 2
lb_vi2 = Label(tk, text="Vitesse init. :", width=10,justify = LEFT,fg = "blue")
lb_vi2.grid(row = 3,column =2)
vv2 = StringVar(tk, value ='20')
IniV2 = Entry(tk,text="vi2", width=3, justify = RIGHT,textvariable=vv2)
IniV2.grid(row = 3,column = 3)
# % profondeur
lb_p = Label(tk, text="'%' prof", width=8,justify = LEFT)
lb_p.grid(row = 4,column = 0)
pctp = StringVar(tk, value ='30')
pct_p = Entry(tk,text="pctp", width=3, justify = RIGHT,textvariable=pctp)
pct_p.grid(row = 4,column = 1)
# % hauteur
lb_h = Label(tk, text="'%' hauteur", width=8,justify = LEFT)
lb_h.grid(row = 4,column = 2)
pcth = StringVar(tk, value ='000')
pct_h = Entry(tk,text="pcth", width=3, justify = RIGHT,textvariable=pcth)
pct_h.grid(row = 4,column = 3)
# Facteur de gravite
lb_gr = Label(tk, text="Gravite", width=8,justify = LEFT)
lb_gr.grid(row = 5,column = 0)
gg = StringVar(tk, value ='3')
Gr = Entry(tk,text="Vi", width=3, justify = RIGHT,textvariable=gg)
Gr.grid(row = 5,column = 1)
def launch_satel():
# diametre mobiles m0, m1 et m2
Phi_m0 = HAUTEUR/20
alph_1 = float(alpha1.get())/100
alph_2 = float(alpha2.get())/100
Phi_m1 = int( Phi_m0 * alph_1**0.5)
Phi_m2 = int( Phi_m0 * alph_2**0.5)
pct_hauteur = float(pct_h.get())/100
pct_profondeur = float(pct_p.get())/100
# position inititale des masses
a = HAUTEUR/2
x0 , y0 = (0,0)
x1 , y1 = (-a,0)
x2 , y2 = (-a*pct_profondeur,pct_hauteur*a*3**0.5/2)
# Position du centre
Gx = (x0 + x1*alph_1 + x2*alph_2) / (1 + alph_1 + alph_2)
Gy = (y0 + y1*alph_1 + y2*alph_2) / (1 + alph_1 + alph_2)
# Calcul de la translation de G au centre
Cx , Cy = ( LARGEUR/2 , HAUTEUR/2 )
tx , ty = ( Cx - Gx , Cy - Gy )
# repositionnement des masses avec G au centre du canevas
x0 , y0 = (x0 + tx , y0 + ty)
x1 , y1 = (x1 + tx , y1 + ty)
x2 , y2 = (x2 + tx , y2 + ty)
ox0 , oy0 = (x0 , y0)
# position des cercles
TL0x = int(x0 - Phi_m0 /2 )
TL0y = int(y0 - Phi_m0 /2 )
BR0x = int(x0 + Phi_m0 /2 )
BR0y = int(y0 + Phi_m0 /2 )
TL1x = int(x1 - Phi_m1 /2 )
TL1y = int(y1 - Phi_m1 /2 )
BR1x = int(x1 + Phi_m1 /2 )
BR1y = int(y1 + Phi_m1 /2 )
TL2x = int(x2 - Phi_m2 /2 )
TL2y = int(y2 - Phi_m2 /2 )
BR2x = int(x2 + Phi_m2 /2 )
BR2y = int(y2 + Phi_m2 /2 )
#vitesses initiales
K =0.1
vitesse1 = K * float(IniV1.get())
vitesse2 = K * float(IniV2.get())
vi1 = -vitesse1
vi2 = -vitesse2
V1x, V1y = (0,vi1)
V2x, V2y = (0,vi2)
# delta t
dt = 0.1
i =0
Gm0 = 500 * float(Gr.get())
# proportion masse 2 / masse 0
main_xaxis = canvas.create_line(0,Cy,LARGEUR,Cy,fill="gray")
main_yaxis = canvas.create_line(Cx,0,Cx,HAUTEUR,fill="gray")
m0 = canvas.create_oval(TL0x,TL0y,BR0x,BR0y,fill="yellow")
m1 = canvas.create_oval(TL1x,TL1y,BR1x,BR1y,fill="red")
m2 = canvas.create_oval(TL2x,TL2y,BR2x,BR2y,fill="blue")
print 'd10sq;d10;G10;G10x;G10y;d12sq;G12;d12;G12x;G12y;V1xx;V1yy;dx1;dy1;d20sq;G20;G20x;G20y;d21sq;G21;d21;G21x;G21y;V2xx;V2yy;dx2;dy2;dx0;dy0'
while i < 20000:
i = i+1
# mouvement de m1
# influence de m0 sur m1
d10sq = (x1-x0)**2 + (y1-y0)**2
G10 = Gm0 / d10sq
d10 = d10sq**0.5
if d10 < Phi_m0 /3 :
d10 = Phi_m0 /3
G10x = -(x1-x0)* G10/ d10
G10y = -(y1-y0)* G10/ d10
# influence de m2 sur m1
d12sq = (x1-x2)**2 + (y1-y2)**2
G12 = Gm0 * alph_2 / d12sq
d12 = d12sq**0.5
if d12 < Phi_m2/3 :
d12 = Phi_m2/3
G12x = -(x1-x2)* G12/ d12
G12y = -(y1-y2)* G12/ d12
# action globale sur m1
V1xx = V1x + (G10x + G12x) * dt
V1yy = V1y + (G10y + G12y) * dt
dx1 = (V1x+V1xx) * dt /2
dy1 = (V1y+V1yy) * dt /2
V1x,V1y = (V1xx, V1yy)
x1 = x1 +dx1
y1 = y1 +dy1
# mouvement de m2
# influence de m0 sur m2
d20sq = (x2-x0)**2 + (y2-y0)**2
G20 = Gm0 / d20sq
d20 = d20sq**0.5
if d20 < Phi_m0 /3 :
d20 = Phi_m0 /3
G20x = -(x2-x0)* G20/ d20
G20y = -(y2-y0)* G20/ d20
# influence de m1 sur m2
d21sq = (x2-x1)**2 + (y2-y1)**2
G21 = Gm0* alph_1 / d21sq
d21 = d21sq**0.5
if d21 < Phi_m1 :
d21 = Phi_m1
G21x = -(x2-x1)* G21/ d21
G21y = -(y2-y1)* G21/ d21
# action sur m2
V2xx = V2x + (G20x + G21x) * dt
V2yy = V2y + (G20y + G21y) * dt
dx2 = (V2x +V2xx) * dt /2
dy2 = (V2y +V2yy) * dt /2
V2x = V2xx
V2y = V2yy
x2 = x2 +dx2
y2 = y2 +dy2
# repositionnement de la masse m0
x0= Cx *(1 + alph_1 + alph_2) - x1*alph_1 - x2*alph_2
y0= Cy *(1 + alph_1 + alph_2) - y1*alph_1 - y2*alph_2
# switch de l'ancienne position
dx0 = x0 -ox0
dy0 = y0 -oy0
# deplacement
canvas.move(m0,dx0,dy0)
canvas.move(m1,dx1,dy1)
canvas.move(m2,dx2,dy2)
# trace trajectoire
l0 = canvas.create_line(ox0,oy0,x0,y0,fill="yellow")
l1 = canvas.create_line(x1-dx1,y1-dy1,x1,y1,fill="red")
l2 = canvas.create_line(x2-dx2,y2-dy2,x2,y2,fill="blue")
ox0,oy0 = (x0,y0)
traceG = str(d10sq) +";" + str(d10) +";" + str(G10) +";" + str(G10x) +";" + str(G10y) +";" + str(d12sq) +";"
traceG = traceG + str(G12) +";" + str(d12) +";" + str(G12x) +";" + str(G12y) +";" + str(V1xx) +";" + str(V1yy) +";"
traceG = traceG + str(dx1) +";" + str(dy1) +";" + str(d20sq) +";" + str(G20) +";" + str(G20x) +";" + str(G20y) +";"
traceG = traceG + str(d21sq) +";" + str(G21) +";" + str(d21) +";" + str(G21x) +";" + str(G21y) +";" + str(V2xx) +";"
traceG = traceG + str(V2yy) +";" + str(dx2) +";" + str(dy2) +";" + str(dx0) +";" + str(dy0)
traceGm = str(i) + ";" +stg.replace(traceG,'.',',')
"""if i >1400 :
print traceGm"""
tk.update()
#time.sleep(0.0002)
launch_btn = Button(tk, text="Launch", width=10, command=launch_satel,activebackground="orange",activeforeground="blue")
launch_btn.grid(row=6,column=1)
tk.mainloop()
not working, so many undefined and unused variables
edit: ok so I kinda got it working, except still many errors occur, and had to change a bit to get it starting. Are they supposed to move smoothly? Or just at fixed intervals?
I loved the Moebius one. SO appropriate.
Interesting. But which ones have dynamic stability? Those without it would multiply errors with each cycle (thus growing error exponentially), starting with rounding errors even if initially tuned as accurately as possible, until the pattern breaks down. Those with it would self correct errors, perhaps with pendulum like resonance, and only break down after the stochastic addition of positive and negative rounding errors (which is virtually muted by high precision numbers) finally modifies the orbit into an unstable pattern. From experience running such simulations, I know that some of the patterns shown are stable, and some unstable, but don't know the solutions for all of them.
the definition of a chaotic system doesn't allow for stable orbits.
I think the only stable one is 1:12~1:20.
All of others looks unstable
@@fakestory1753 All of them are stable, thats the point. What @C13glass is asking is if any of these are attractors (i.e. if you change the starting condition a very small amount will they move towards the stable configuration shown). I believe there is no stable periodic solution to the three body problem that is an attractor.
@@LilithLuz2 You confused the terms attractor with those simple attractors.
What describe the chaotic system is called strange attractor(but having a attactor isn't necessary).
@@fakestory1753 no, an attractor can refer to any attracting set. The term strange attractor comes from the fact its a dense set in space in which some chaotic dynamical systems eventually reside, but there are plenty of other types of attractors. In particular, single points can be attractors (attracting fixed points), as can cycles as shown in these videos. You may want to look more into the ideas of stable and unstable manifolds.
Some of these make me wonder how they were discovered
This illustrates the incredibly profound complexity that's weaved into the nature of everything we interact with on a daily basis... Fantastic video!
just finished the first book 👽
Very cool. These are plotted in two dimensions, so finding solutions for these is sort of working backwards from the solution, isn't it? Also it would be cool to see the libration points mapped in the animations. Thanks for doing this.
Libration points are a completely different thing. NA for these orbits shown here.
Any 3 points define a unique plane. Since all the bodies are on the same plane, all gravitational forces are on the same plane. Without starting momentum or outside forces, the bodies cannot leave their original plane, so plotting a 3 body problem in 2 dimensions makes perfect sense
@@52flyingbicycles the bodies have an initial speed that could be outside of the plan they define.
the trajectories are beautifully symmetrical
0:05 yellow and geen crash right into each other if they were planets I can assure you there would be no body problem
1000000 body problem*
They are meant to be dots
Try that in 3 D
That must be a mess.
It doesn't really matter since any three points belong to a single plane so they would never leave it and it would look exactly the same when viewed perpendicularity to said plane
@@davideranieri5553 That's not necessarily true. You're assuming that the starting velocities of each object also lie within the plane that the 3 objects form which does not have to be the case in 3D. As an example you could totally have 2 objects orbiting each other in a circle while the third one orbits both inner objects in a circle perpendicular to the other circle. Now whether or not that is a stable orbit is a different question.
@@ReconFX You're right, I was assuming they started still lol
--Some of these seem like variants on the "binary but make one of the stars a smaller binary" and "massive star with two planets in a horseshoe orbit" patterns. The horseshoe is physical, and the multi-level binary kind of thing is stable at large distances and can have orbital resonances. The variations are wacky!
--The one with a big ellipse and a smaller one is totally new to me and has a nice symmetry.
--I've seen the figure-8 before and still don't get it rip, but it's clearly got a nice symmetry to it.
--The one with 3 ellipses where you start with your distance and velocity vectors in an equilateral triangle :D I can imagine varying it to get hyperbolae if you go up to some kind of escape velocity, and a perfect circular orbit if you get initial conditions just right.
--The boring option would be to have something small in a Lagrange point between 2 massive bodies where you adjust the ratio of orbital periods to make them periodic.
Just beautiful
This is beautifull!
Thanks! And, always good to hear from a lab alumnus.
I couldn’t entirely like, because you have used explicit time integrator (here MATLAB’s ode45 or an explicit Runge-Kutta (4,5) formula.) Their problem is unintentional numerical artefacts (incl. numerical chaos, periodicities and other instabilities). While it is first solver to try, then you should switch implicit time integrators (e.g., ode23tb or ode15i) for superior numerical stability, needed especially in such long-duration simulations.
nice, but are these solutions stabil.
is the solution still similar, when there is a little disturbition?
will it result in these motions, even started at a little different point or velocity?
How to these react to small disturbance? Are some stable when touched?
Can you solve in 3D with the bodies free to travel above and below the plane in the Z axis?
3:31 I wonder something similar to this would with any number of stars of equal mass
Yes by symmetry I think it does : if you consider one of the stars, then the center of mass of all the others is necessarily along the radial direction (by symmetry). Therefore this one star follows a standard Keplerian orbit. And by symmetry, this is valid for all the stars.
See: "Globular Cluster" As the masses of the individual particles go down relative to the total, the system becomes more stable.
Here's the video on surface knots:
czcams.com/video/h68AxRiGxBw/video.html
- i will try to figure out a way to turn the orbits into knots this week. Also, Keenan crane does a good series on curvature flows - these orbits remind me (ironically) of repulsive curves:
czcams.com/video/-uXFYpVumh4/video.html
- you would need a minimizer that minimizes the action while avoiding collision, but i think that would fall out naturally with parameterization over proper time
- because the knots are knots, the un-knotting minimizer will generate smooth non-intersecting curves that are also valid orbits rather than separate loops
- all of this would need to take place in some kind of complex, conformal or projective time axis
Edit: you may be asking yourself, my solutions are numerical approximations - but from a discrete differential geometry approach your solutions may be discrete surfaces and it is often possible to recover a continuous solution from a discrete one.
Consider the simplest case of a two-body orbit. From the barycenter lab frame, you have two infinitely long open cylinders vertical in the lab time axis, one inside the other. Assign a phase to every point on each cylinder and picture it using a color wheel - you have a helix around both cylinders that are out of phase by i.pi modified by a factor proportional to the distance between the cylinder walls. The lab frame is frame-dragged requiring an additional complex factor. For a frame on any of the bodies, use a Mobius transform of the entire system. To take a single orbit solution, take a slice through the complex plane = choose a phase = select a starting angle for the two bodies relative to the lab frame. Then project down onto the 2d plane.
For unstable orbits, the surfaces intersect (including in the complex axis! See: 4d surface knots) or diverge. It would be interesting to see what projection allows for taking planar slices like you can with conic sections...
Enjoyed this !!!!!!
Excellent! Did you find these periodic solutions by giving them some random initial conditions and let them luckily converge to these? Anyway, amazing!
it’s chaotic, so probably no
And which of them are actually stable?
I can't help but wonder if these are sections through embedded knotted surfaces
Hey ! I don't know if you will reply since you posted it a long time ago but what is the method to find such periodic orbit ?
And also are some of these orbits stable ?
What do we do when two bodies collide? Do we use the theory of elastic collisions? Do we consider those 3 bodies as material points or extended bodies?
I'm fairly certain collisions are ignored in this.
I guess we have to assume that the three bodies are stars, and that they will be planetless. TBH, even if the stars started out with planets, a few cycles in on any of these orbital patterns, those planets will be either kicked out of the trinary system, or else thrown into one star or another. Even if they did somehow survive, they'd be uninhabitable by our sort of life (until its disproved, I'm hoping there's more than terrestrial kinds out there), and probably most other sorts, too. As for sapient life, well, given that it requires relatively stable environments to evolve, any of these periodic orbital patterns are non-starters.
very interesting.
Hi i am also working on this problem, can you give me initial conditions for a few of those solutions?
Life on green or yellow in No. 4 would probably suck.
You should publish this!
Imagine living on a planet around such triple star system
Proxima b
If you plot in a 3D graph the x coordinate of each color, then find the intersection cubes exhibiting periodicity and highlight them, I wonder what that 3D graph will look, composited over the same 3D graph for the y and z coordinates. Perhaps color-coding highlights for which graph they are from.
I suspect the 15 orbital configurations here are all planar, 2-D
i am also working on this problem but i am facing one weird issue, when my particles gey very close , the simulations "breaks" and shows solid 2D triangular planes instead of lines as a path of the particles , but if i decrease the time interval of each interation this problem goes away but then calculations cost more time, is there any way to tackle this issue?
you could try using a dynamic step size. so make the step size smaller when the velocities of the particles are large or if they get close together, and make it larger when they're further apart and moving slower
Or change the step size based on some estimated error. Higher order and implicit ode solvers may also help, but the adaptive stepsize is an absolute must.
Every time there's a closest approach between two bodies I hear a note/beat. I wonder if this could lend itself to making some interesting music.
But supposing those are stars how many and which of these solutions would allow for a planet to orbit around one of the bodies without getting destroyed or suffering extreme weather conditions?
Could tou kindly send over the initial conditions for these periodic orbits?
Wonderful! Are any of these stable under perturbation?
Exactly my question also... ;)
Probably not too stable
@@Anyo92 They either are or not.
@@jackmclane1826 depends on amount of perturbation
Great video! I am really curious what will happen to a planet in this system?
Impresionante
Imagine someone having to make a physical model for some of these systems
How to know il this solutions are stable or not ?
Извините за русский, стабильных сихстем N количество, но всех систем М количество, множество N входит во множество М. Но проблема в том, что нет формулы которая могла бы просчитать любую из этих систем. Она должна нам показать перспективу траэкторий движения тел.
Are they all confined to the plane?
Are all these solutions planar, existing in only a 2-dim plane?
if the three bodies start coplanar there's no reason for them to leave that plane
I agree that there are no stable solutions for astronomical objects in our universe for the n-body problem. Nevertheless a mathematically stable solution could be (nearly) realized for millions or billions of years somewhere out there.
a lot of these look like the stars are coliding but I know that's just the ilusion created by a scale model. The stars are just passing by each other at an astronomically close distance like a few tousand miles.
1:17 This is the solution for the Alpha Centauri system.
I know that in three-body motion, the center of mass does not change due to the law of conservation of momentum. Can I ask why the center of mass moves in the first video?
Perfectly Balanced
As All Things Should Be
It's definitely interesting, but some of the solutions here have two bodies colliding or passing through each other
Cixin brought me here, too
Nice video and I appreciate it
Initial conditions are so subtle
Now I understand that 3 suns are impossible in our universe
What the pattern at 1:12 called?
flower pattern by trisolaries
@@arisoda thx
Would people feel those close encounters?
soo cool
Dehydrate
Your first simulation is wrong: there is a move of the center of mass, which means you have to reduce the dt or dx/dy in your simulation.
Beautiful videos! How did you search for and find these configurations? There are so many possible initial conditions.
Thanks! This method is pretty dumb. it runs a multivariable numerical function solver (FSOLVE) on top of the simulation in order to find initial conditions and simulation time which result in [final conditions] - [initial conditions] = [zeros]. Seeding FSOLVE with different initial guesses causes the method to be sucked towards the closest local solution. Since this is pretty much just for fun, I had my computer pick lots of random initial conditions overnight and picked the ones that actually converged. Definitely not rigorous, but still interesting!
Matthew Sheen: Cool... and doesn't sound dumb to me ☺️. Thanks for sharing. You might get a kick out of the sci-fi novel "The Three-Body Problem", which deals with this among other things. That novel got me searching CZcams and finding your video.
I'll check it out. Thanks for the suggestion!
Would you feel the acceleration on a planet periodically sped up like those ?
2:40 - 1 BED 2 AMONGUS
The fact is that they have 3 suns and a planet that they can live on. But they have to find a stable system that their planet can have a stable temperature and not be influenced by the three suns' gravity.
Raymond Tan what
@@piaresquare5424 You're suggesting a 4 body problem and you're not wrong. Cixin seems to suggest that since the influence of the trisolaran planet is negligible in the system as a whole due to it's small size, we should ignore it. But wereally can't - in an unstable system, minor variations amplify non deterministically. In any case, Alpha Centuri is a bad example as Proxima C. is very loosely bound to Alpha Centuri AB system and will unlikely cause the kind of issues for trisolarans.
yeah wow, cos the universe is not sensitive to initial conditions, and is typically restricted...
Great use of a solver software
Mohanraj Jk why are you so butthurt lol?
The 2nd one is actually a 2 body system
Imagine trying to figure out how space works on a planet with this kind of orbit
Well the trisolarans did it ;)
@@baraapudding I read that story and had trouble getting past the notion that life could ever evolve there to begin with, let alone such complex life. Still an interesting story though I guess.
Awesome! How hard is it to work out our solar system then? Have you ever considered our universe is misdescribed?
The planets don't really affect each other by a lot, since they are outside each other's sphere on influence. The only major gravitational "force" they experience is the gravitation of the Sun.
@@DelayRGC ummm the sphere of influence for gravity is infinite. And the perturbation due to other planets is easily measured. It was measurable hundreds of years ago
@@zoltankurti The SOI is defined as the region of space in which one body's gravitational influence is dominant. It's a slight simplification of a different concept, (the Hill sphere) but it's adequate in most cases.
Of course I am aware of the fact that gravity extends to infinity, but gravity decreases inversely propertional to the square of the distance.
As an example, Io experiences a much greater gravitational force from Jupiter as it does from the Sun. That's why it can orbit around Jupiter.
But as orbital radius increases, Jupiter's gravity quickly decreases and the orbit becomes a heliocentric one. The (simplified) boundary between these two behaviors is the sphere of influence.
@@DelayRGC all right, got it. I wasn't sure how precise you wanted to be. For some people the gravitational influence of planets on eachother is very significant. It's a bigger effect than general relativity even for Mercury, where "GR is the strongest". So in some areas this is incredibly important, that's where I was coming from. Until you know with low relative error how planets affect eachother you have basically no information about how GR affects orbits.
@@zoltankurti Ellipses around the Sun are good enough when describing the basic properties of the Solar System. You can integrate if you want to, but the difference between (GR,) Newton and Kepler is very small most of the time.
Awesome animation ; just a question, are we sure these are indeed periodic solutions and not consequences of the inevitable discretisation of the problem for the sake of simulation ?
Planet: I will perturb this solar system.
im surprised some of them arent symmetrical
Most of the patterns of their orbits are symmetric, while only four of them are not.
I wonder if there's any planetary system in the Universe that follows any of these orbits.
They are unstable: any small deviation from the solution will lead back to chaos. Because of this, no such configuration can naturally occur.
okay, does Keppler's third law hold in here?
I would think it doesn't, at least without considering something like, the projection of the center of mass in the system
Kepler's laws only applies to the 2 body problem, or cases that can be well approximated by a 2 body problem (such as the motion of moons around planets and planets around stars, but not stars around the galaxy for example). Maybe you could extrapolate some Kepler's law analogs for stable 3 body configurations.
Everyone on those planets is throwing up
Wow! You just stopped me from deleting the CZcams app!
So........the crop circles are messages....From THEM.
shit we need to start simulating based on the crop circle shapes!
2:59 green is sus - AMOGUS
The universe has more patterns that we can't recognize than those that we can. AI will recognize and share a few more with us. Until it decides to not.
So our solar system is currently under the influence of a periodic solution?
I would love to know and see if there are stable planetsystems in such interesting pathes 😅
Look at the first one
Есть не толко проблема расчёта системы, но и есть проблема расчёта каждой из тел этой системы.
Also assuming a radius value for these fake planets just like earth
2:56 amongus
2:47 amogus
三体？
but in the real universe, they can't collide...
3:01 sus
If you would draw the "field" as a background map, you will see the general solution. The lagrange points. the potential and drive the stable paths from that.
Any 3 closed paths that do not intersect will give you new solution.
It's not as simple as that. The trajectory of each object depends on that of the others, so you can't just find closed paths independently and add them up. Also, Lagrange points become useless when the third body has a mass on the same order of magnitude as the other two.
2:55 is sus
Okay, now do it in 3-dimensional space. XD
Should be easy seeing as that's supposedly whats happening in observable reality of the night sky but you never will see this as s it's bs
3:33 this is not a solution, play it for long enough and the simulation becomes unstable. Such I suspect with the other "solutions"
How do you prove that it is not a stable solution ?
@@Gachette00 you run it for long enough. the examples in this video would become stable after a few hundred orbits at least. Even three body orbits like the moon, earth and sun are unstable after a few billion years
@@incription but how do you know exactly the initial conditions of the orbit ?
Or is there a proof that all the orbits which look like that are unstable ?
Because I have heard that some orbits, like the "infinite" one, had been proven to be stable.
Don't worry, I know that earth-moon-sun orbits are unstable.
The stable orbits are almost impossible in the universe, they are only theorical.
@@Gachette00 to clarify, there are mathematical closed solutions to specific orbits, in which their positions at time T can be calculated by a function. You are correct that in reality, there is no infinitely stable orbit
@@incription to specific orbit there is also some mathematical exact periodic solutions.
They talk about them in the Wikipedia page, in the specific solution paragraph
en.m.wikipedia.org/wiki/Three-body_problem
So I don't know why the one that is shown couldn't be a solution to the problem. (Even if it can't exist in the physical world)
it's to agressive for my eyes, Im used to dark mode (Its not a joke)
Всё это интересно. .но. сдается мне вы не учитываете скорость распространения взаимодействия .а именно скорость света
They exist in two dimensions. But not in three!!!!
anything that exists in two dimension also exist in three, so all of those are valid 3D solutions. It would be very insteresting to look at strictly 3D solutions, but they're probably harder to find.
But these solutions only exist in some ideal world that doesn't exist. So not exactly real solutions.