As the default simulation played out beautifully on-load, I immediately started to question: "hang on, I thought there wasn't a solution for 3 bodies, but this looks stable".
Before I could complete the thought, it fell apart magnificently :)
The default configuration is a special case. They are all in a stable orbit around the common barycenter, always forming an equilateral triangle. We actually have closed form solutions for this kind of configuration.
These are meta-stable, in that only a perfect initial state (positions and velocities) will be stable. Even then, I expect quantum uncertainties would kick in at some point. (In the simulator, the default system goes unstable at about 500 seconds, probably due to the limits of floating point math.)
It's not obvious from the UI but you can enter small mass changes and watch things slowly fall apart. E.g. 1.0001 will work even though the UI displays 1.0 after you hit enter.
Love this, reminds me of a Windows program (whose name I’ve forgotten) that I was playing with some decades ago… Solarwinds or something similar. You could add planets/masses and play with orbits, trajectories and all sort of options.
Seems like there is no way to actually make those collide with each other. Even when they are really close, they are just make a pass go in other direction.
Just for fun, I set it up with six bodies, each 1 unit mass and at 1 unit from origin, but along the three different axes (an octahedron). No initial velocities. Start the sim, they fall towards the center, and then BLAMMO they rocket off in opposite directions at high velocity. Clearly, no conservation of energy here (at least when the bodies are arbitrarily close to each other).
Here is a thought; Instead of using F=ma, use the equations of motion from GR:
================================================================================
SCHWARZSCHILD METRIC AND GEODESIC EQUATIONS OF MOTION (SUMMARY)
================================================================================
I. THE SCHWARZSCHILD METRIC (g_uv)
The spacetime geometry is defined by the *line element*, ds^2, which relates coordinate changes (dt, dr, d(phi), etc.) to physical distance or proper time:
ds^2 = g_uv * dx^u * dx^v
For the Schwarzschild vacuum solution, the line element in the equatorial plane (theta = pi/2) is:
ds^2 = -(1 - r_s / r) * c^2 * dt^2 + (1 - r_s / r)^(-1) * dr^2 + r^2 * d(phi)^2
The Lagrangian L for the geodesic path is constructed directly from the metric:
L = (1/2) * [ g_tt * (dt/d(lambda))^2 + g_rr * (dr/d(lambda))^2 + g_phiphi (d(phi)/d(lambda))^2 ]
II. CONSERVATION LAWS (FROM EULER-LAGRANGE EQUATIONS)
A. TIME EOM (Conserved Energy E)
Since the metric is time-independent, the quantity conjugate to t is conserved: *Specific Energy (E)*.
EQUATION (1): Time Evolution
d(t)/d(lambda) = E / ( c^2 * (1 - r_s / r) )
B. PHI EOM (Conserved Angular Momentum L_z)
Since the metric is symmetric with respect to phi, the quantity conjugate to phi is conserved: *Specific Angular Momentum (L_z)*.
III. RADIAL EQUATION OF MOTION (FROM THE METRIC CONSTRAINT)
The radial EOM is derived by imposing the metric normalization condition (g_uv * u^u * u^v = epsilon).
A. MASSIVE PARTICLES (Mass m > 0)
The proper time (tau) is the affine parameter (lambda=tau), and the normalization is epsilon = c^2.
The final EOM is:
(dr/d(tau))^2 = E^2/c^2 - V_eff^2
The motion of any particle (massive or massless) in the Schwarzschild spacetime is determined by the following three coupled first-order differential equations:
A. TIME EVOLUTION:
d(t)/d(lambda) = E / ( c^2 * (1 - r_s / r) )
B. ANGULAR EVOLUTION:
d(phi)/d(lambda) = L_z / r^2
As the default simulation played out beautifully on-load, I immediately started to question: "hang on, I thought there wasn't a solution for 3 bodies, but this looks stable".
Before I could complete the thought, it fell apart magnificently :)
The default configuration is a special case. They are all in a stable orbit around the common barycenter, always forming an equilateral triangle. We actually have closed form solutions for this kind of configuration.
See https://en.wikipedia.org/wiki/Three-body_problem#General_sol...
PS: The site has more stable presets under load preset.
These are meta-stable, in that only a perfect initial state (positions and velocities) will be stable. Even then, I expect quantum uncertainties would kick in at some point. (In the simulator, the default system goes unstable at about 500 seconds, probably due to the limits of floating point math.)
>probably due to the limits of floating point math
It's due to the integration scheme (2nd order, albeit symplectic) and time step (5e-4, ok if better scheme is used).
There are quite a few solutions ;)
https://www.sciencealert.com/we-just-got-12000-new-solutions...
Discussed before here:
https://news.ycombinator.com/item?id=45967079 (245 and 112 comments)
Love this.
It's not obvious from the UI but you can enter small mass changes and watch things slowly fall apart. E.g. 1.0001 will work even though the UI displays 1.0 after you hit enter.
Love this, reminds me of a Windows program (whose name I’ve forgotten) that I was playing with some decades ago… Solarwinds or something similar. You could add planets/masses and play with orbits, trajectories and all sort of options.
Seems like there is no way to actually make those collide with each other. Even when they are really close, they are just make a pass go in other direction.
Just for fun, I set it up with six bodies, each 1 unit mass and at 1 unit from origin, but along the three different axes (an octahedron). No initial velocities. Start the sim, they fall towards the center, and then BLAMMO they rocket off in opposite directions at high velocity. Clearly, no conservation of energy here (at least when the bodies are arbitrarily close to each other).
Simple pleasures.
I doubt they implemented any collision physics.
https://youtu.be/ByLhzd5biag
Commentary on close radius interactions. Very interesting wrt nuclear forces!
Love the presets! The Broucke one is my favourite
Wow, this looks really nice on a 240 Hz display.
Would be a great screensaver!
Here is a thought; Instead of using F=ma, use the equations of motion from GR:
================================================================================ SCHWARZSCHILD METRIC AND GEODESIC EQUATIONS OF MOTION (SUMMARY) ================================================================================
I. THE SCHWARZSCHILD METRIC (g_uv)
The spacetime geometry is defined by the *line element*, ds^2, which relates coordinate changes (dt, dr, d(phi), etc.) to physical distance or proper time: ds^2 = g_uv * dx^u * dx^v
For the Schwarzschild vacuum solution, the line element in the equatorial plane (theta = pi/2) is: ds^2 = -(1 - r_s / r) * c^2 * dt^2 + (1 - r_s / r)^(-1) * dr^2 + r^2 * d(phi)^2
The corresponding non-zero metric components (g_uv) are: g_tt = -(1 - r_s / r) * c^2 g_rr = 1 / (1 - r_s / r) g_phiphi = r^2
Where: r_s = 2 G * M / c^2 (Schwarzschild Radius)
The Lagrangian L for the geodesic path is constructed directly from the metric: L = (1/2) * [ g_tt * (dt/d(lambda))^2 + g_rr * (dr/d(lambda))^2 + g_phiphi (d(phi)/d(lambda))^2 ]
--------------------------------------------------------------------------------
II. CONSERVATION LAWS (FROM EULER-LAGRANGE EQUATIONS)
A. TIME EOM (Conserved Energy E) Since the metric is time-independent, the quantity conjugate to t is conserved: *Specific Energy (E)*.
EQUATION (1): Time Evolution d(t)/d(lambda) = E / ( c^2 * (1 - r_s / r) )
B. PHI EOM (Conserved Angular Momentum L_z) Since the metric is symmetric with respect to phi, the quantity conjugate to phi is conserved: *Specific Angular Momentum (L_z)*.
EQUATION (2): Angular Evolution d(phi)/d(lambda) = L_z / r^2
--------------------------------------------------------------------------------
III. RADIAL EQUATION OF MOTION (FROM THE METRIC CONSTRAINT)
The radial EOM is derived by imposing the metric normalization condition (g_uv * u^u * u^v = epsilon).
A. MASSIVE PARTICLES (Mass m > 0) The proper time (tau) is the affine parameter (lambda=tau), and the normalization is epsilon = c^2. The final EOM is: (dr/d(tau))^2 = E^2/c^2 - V_eff^2
EQUATION (3M): Radial EOM (Massive) (dr/d(tau))^2 = E^2/c^2 - c^2 * (1 - r_s/r) * ( 1 + L_z^2 / (c^2 * r^2) )
B. MASSLESS PARTICLES (Mass m = 0) The normalization is epsilon = 0. The final EOM is: (dr/d(lambda))^2 = E^2 - V_eff^2
EQUATION (3P): Radial EOM (Massless / Photon) (dr/d(lambda))^2 = E^2 - (1 - r_s/r) * L_z^2 / r^2
--------------------------------------------------------------------------------
IV. SUMMARY OF GEODESIC EQUATIONS OF MOTION (EOM)
The motion of any particle (massive or massless) in the Schwarzschild spacetime is determined by the following three coupled first-order differential equations:
A. TIME EVOLUTION: d(t)/d(lambda) = E / ( c^2 * (1 - r_s / r) )
B. ANGULAR EVOLUTION: d(phi)/d(lambda) = L_z / r^2
C. RADIAL EVOLUTION (Specific): 1. Massive Particle (using d(tau)): (dr/d(tau))^2 = E^2/c^2 - c^2 * (1 - r_s/r) * ( 1 + L_z^2 / (c^2 * r^2) )
2. Massless Particle (using d(lambda)): (dr/d(lambda))^2 = E^2 - (1 - r_s/r) * L_z^2 / r^2
================================================================================
This also holds for a non-rotating black hole.
Is this AI generated?
Yes, just took a few seconds.
It is of course a very well known result.
Not sure why all the down votes.