A rudimentary simulation of the three-body problem

It's a hundred times less polished than rebound (that readme looks seriously cool!), but in 2021 I also wanted to toy around with orbital mechanics / gravity and couldn't find a quick and easy simulator so I made this to run in a browser: https://lucgommans.nl/p/badgravity/

Since it's so rough on the edges (especially on mobile, initially I was surprised it works at all), here's the steps for the mentioned example of making Jupiter 10x heavier:

    1. Open the scenarios on the left and click play on the inner solar system to load that up
    2. Click the plus on the outer planets to add them in (if it looks like nothing happened: zoom out. Space is big and this is to scale)
    3. Fold out the "bodies" section and alter the mass for "J"upiter. The change is applied live.
    4. Optionally press Restart to restart with the current settings but back at their initial positions and speeds

The help/about page (https://lucgommans.nl/p/badgravity/about.html) contains links to all other orbit projects I could find. Seeing Rebound as well as the OP, I should probably add a "libraries" section! Or do you think that should just go with "Software to download" alongside Stellarium and such?

Looks awesome. I have a question that might be pedantic but I think you could speak to. Coming from an engineering mindset, I like the use of the term 'integrator' instead of 'solver'. In MATLAB, 'solvers' are used to iterate states of ODE models. But the term 'integrator' is more intuitive to me. Can you speak to the use of one term vs another in the ephemeris community?

How is it random if the 3 body problems are chosen to have overlapping ellipses that look beautiful

I love this bot, thanks for sharing!

‘Strange attractors’: https://en.wikipedia.org/wiki/Attractor

“An attractor is called strange if it has a fractal structure.”

Looks like fluid dynamics / turbulence to me

It wasn't the first program I wrote, or even a program I wrote, but in middle school a friend wrote a 3-body integrator in BASIC (sun, earth, moon). That single 20 line program shaped my entire world view for a long time (decades), implying to me that we could, if we had powerful enough computers, simulate all sorts of things... even entire universes (which was also an idea that I explored with cellular automata).

It's not a particularly helpful worldview and can often be harmful if you're working with complex systems, or systems that require more than o(n(log(n)) per step, or any number of other real-world problems scientists face.

Many years later I was impressed at how well astronomy packages work (IE, "it's time T at local L, what is the angle of the sun in the sky?") and stumbled across this paper by Sussman: https://web.mit.edu/wisdom/www/ss-chaos.pdf which shows some pretty serious work on future prediction of solar system objects.

I think symplectic integrators are typically used, which are derived from hamiltonian mechanics

https://en.wikipedia.org/wiki/Symplectic_integrator

It's true that Euler integration is about as crude as you can get, but you don't need to reach to Lagrangians for improvement; something like Verlet integration can already bring dramatic gains with fairly small changes needed.

Can convert Euler's method to a symplectic integrator utilizing v_{n+1} when computing x_{n+1}. That said although such integrators are widely used (usually of higher order than Euler's) in celestial mechanics, one is not restricted to them. For example Bulirsch-Stoer is also very used even if it isn't symplectic because remains accurate (energy error very low) even on long integrations.

Would it make sense to explicitly implement conservation of energy?

I.e. do a simple method but calculate the total energy at the beginning, and at each step adjust the speeds (e.g. proportionally) so that the total energy matches the initial value - you'll still always get some difference due to numerical accuracy issues, but that difference won't be growing over time.

This is irrelevant to the unsolvability.

That the three body problem is unstable and that no analytic solution exists are completely independent statements.

The upright pendulum is also an unstable ODE, yet it has an analytical solution.

This only demonstrates that the system is chaotic, not that there is no closed form solution.

A few years ago a friend and I made something similar to universe sandbox, though only with the gravitational simulation part: https://github.com/fayalalebrun/Astraria

Surprisingly enough the jar still runs without issue. Something which probably would not be the case for linux binaries, but maybe for windows.

Not to forget https://www.kerbalspaceprogram.com

It's ok, I'll just make sure never to measure the CMB for variations.

N-body problems can't be analytically solved. However, you can still compute integrals into the future (with some acceptable error), you just need to step through all the intermediate states along the way

In the case of the solar system, yes, it helps that the Sun is much more massive than everything else (and then Jupiter is 4 times more massive than Saturn, the next biggest) - you can go a long way to a "reasonable" solution by starting with the 2-body solution if only the Sun affected each planet, and then adding in the perturbation caused by Jupiter and Saturn. (In fact, that's how we predicted the existence of Neptune, by noticing that there were extra perturbations on Uranus beyond those, and hence another massive planet must exist, far enough away from the sun to only significantly affect Uranus).

Predictions of the solar system state are accurate only on "short" periods. The solar system is chaotic, and predicting its state after few million years is no more possible that predicting the weather for next year. This does not preclude making very accurate on short period.

Note that n-body problems are not particularly complex or hard to solve compared to other chaotic systems. In many ways, the existence or nonexistence of closed-form solutions is mostly a distraction: it merely reflects our choice of primitive functions and there is no sets of primitive functions that is stable for addition, multiplication, composition, inversion and integration. Typically, even the simple integral ∫ eˣ/x dx cannot be decomposed into more elementary functions.

But that doesn't matter in practice, because we are already using numerical approximation to compute primitive functions that are not implemented in hardware. Using numerical solvers to compute solutions to ODE is not so different. A good illustration of that point is that there is an analytic solution for the 3-body problem (in the form of an infinite series in t^{1/3}). But this solution is useless for computing orbits because it has bad convergence properties. In other words, it is better to use a numerical solver rather than stick to the analytical solution. And a similar phenomenon exists for polynomial equation of degrees 3 and 4: the exact formula is numerically unstable, and its better to use a numerical solver when one wants a numerical solution.

IAMAP, but n-body problems result in non-linear partial differential equations, to which only a few special cases are known, and even that case its do to simplifications, e.g., treating planets as point masses and ignoring tidal forces, or ignoring the pull of the planets on the Sun, (or on each other).

One such case where a solution is known is the Lagrange point of the Earth-Moon-Sun system (and similiarly for other points) https://en.wikipedia.org/wiki/Lagrange_point But in reality, they exist only as an approximation. They aren't truly stable.

My understanding the way to calculate spacecraft, asteroid, etc. trajectories is just through a discrete simulation.

Like f you don't know how to solve the antiderivative of a given function, you can still calculate the integral since you know the value of the function.

To add to the points from others: 3+-body systems are generally chaotic, so you cannot predict arbitrarily far into the future, but the solar system is reasonably well-behaved in that manner, so the timescales are long, but we don't know where the planets in the inner solar system will be in their orbit in ~5-10 million years (as in, that's the timescale where the error bars for the position in the orbit span the whole orbit). Of course, if you care about more precise predictions then the timescales are shorter: eclipse predictions more than 1000 years in the future are likely to be quite inaccurate.

For almost any differential equation there is no analytical solution for it's initial value problem. That the n-Body problem behaves that way is unsurprising and poses no inherent challenge to making predictions.

Computers can easily solve initial value problems for most ordinary differential equations. They integrate them, calculating an approximate solution after every small, but finite step.

Getting an approximate solution to the 3-Body problem can be achieved in around 20 lines of python, without having to use any libraries. It is a remarkable simple and effective technique.

You can solve it numerically using Finite Differences. Basically using linear approximations

Loved your website. Thanks for sharing.

See https://en.wikipedia.org/wiki/Adaptive_step_size

The book is mis-named. Their system was not 3 bodies, but 4 (3 stars, plus a planet). And the system is so chaotic that even that little planet will make a huge difference over time. And even beyond that, the bodies themselves were transformed, having the ability to tear atmosphere away from each other.

I was wondering about this too, but I think they're not colliding. I think they're pulling towards each other and getting tightly pulled around each other so they end up slingshotting and flying back the way they came, in a way that looks like a bounce.

You can see this in the very beginning of the simulation, with the blue and green dot.

Can anyone say if this is actually accurate? It seems like an unintuitive motion to me, but I'm often surprised by how these things work.

Luckily??! Could have saved everyone a lot of trouble and kept them in the proper number of dimensions.

Should a collision conduct to a simulation of a 4 body problem ?

that might just be because of my low resolution gif ;)

isn’t that story a 4-body problem?

Probably a misnomer. And issue isn't really about the planet affecting the movement of the 3 suns (its mass can be considered deligible compared to them) but how the suns affect the movement of the planet. Essentially they're looking for a solution to the restricted 4bp rather the 3bp.

No, it's just three. The reason it's a thing – as far as I understand – is that solving analytically for one body is trivial: it's stationary. Solving analytically for two bodies is possible but takes some calculus. Three bodies have chaotic behaviour and need to be simulated.

There’s a saying in mathematics circles which I’ll butcher here: “everything is either a linear system, reducible to linear systems, or unapproachable.”

Think about how bad we are at analytically solving “simple looking” diff-eqs and the above statement starts to sound too true.

This isn't planet level, but shows surface: https://universesandbox.com/

Maybe its because I made an honest effort of getting a PhD in physics, but I absolutely do not understand this perspective.

Like yes, we have a really hard time talking about just about anything as finite object with physical extent, but jokes about frictionless spherical cows moving in simple harmonic motion started in secondary school. The gaps and shortcomings should not come as a surprise. But most of us also hold devices in our pockets that leverage actual quantum phenomena to function at all (diodes of any stripe only work because of quantum transitions). So while its true that there are a variety of unsolved and potentially unsolvable problems in physics, its a gross misunderstanding to say that it can barely answer simple questions.

I think about the Born-Oppenheimer approximation a lot, as its so obviously a hack to even do the math at all, but it undergirds basically all of solid state physics.

What in this post isn't honest?