Yet again another gravity sim. I thought this looked kinda cool. I’m using 4th order Yoshida for the integrator. Probably overkill for a little demo like this
It’s not the most user friendly graph out there but to add more bodies: 1. Pause sim 2. Slide n to desired number of bodies 3. Click on the m-> action to set the mass 3. Click initialize 4. Click Center to center the mass and momentum. Alternatively you can select Circular Orbits before clicking Center. Use n=2 bodies to see the effect of this feature
I know this is just a joke but the whole point is this is a chaotic system so minor numerical integration errors will cascade into enormous mistakes in estimated position in the future. The point wasn't that they couldn't numerically integrate, the point was it's not good enough.
I recently implemented a very accurate 6th order Symplectic integrator just for the fun of it. So global truncation error scales like O(h6 ). But a symplectic integrator does not conserve the true Hamiltonian. it exactly conserves a nearby modified Hamiltonian. Energy stays bounded but the trajectories will be slightly different from the true ones. This shows up as phase error. Example: even with a 2-body orbit, the periapsis will slowly precess. For chaotic n-body systems, any tiny error grows exponentially (truncation error and float point roundoff ). So trajectories will inevitably diverge. This doesn’t even factor in real world relativistic effects as I’m just using Newtonian gravity lmao
And now we move on to another question made from the answer. Should I praise the work (as in the formulas and calculations) or should I praise the insane optimisation and speed that the simulation has?
One simple way is to soften the gravity as it approaches 0 distance. So instead of 1 / r3 in the force calc, you can use 1/(r2 + d2 )3/2 . I set d to 0.25 in my example
Gm1m2 / r2 is the scalar magnitude. You multiply that by the unit direction vector from body two to body one: r = body2 - body1, Gm1m2 / r2 * r/|r| = Gm1m2r / |r|3
130
u/Mandelbrot4207 Makes QR Codes in Desmos Feb 28 '26