N-body simulation with Jacobi coordinates

The gravitational N-body system simulated using Jacobi coordinates in the Hamiltonian formalism.
Dynamics are presented for various planar configurations. The Jacobi coordinates are depicted as green spheres connected with purple rods, which should be interpreted as the vectors of canonical coordinates governed by Hamilton's equations.

0:00 intro
0:14 introducing jacobi coordinates. Unstable system
0:29 figure eight (three planetary bodies)
0:36 figure eight orbiting two bodies. Stable system*
1:00 Broucke-Henon orbiting a celestial body. Stable system*

*conjectured based on long-term simulation

The Hamiltonian system is expressed explicitly without approximations. The system barycenter is fixed at the origin with zero momentum of the entire system.

This specific implementation uses the Hamiltonian originally presented by Wisdom & Holman (1991) DOI:10.1086/115978 but without modifying the system and without application of specialized integration algorithms. The dynamical behavior of the system is identical to that of Cartesian coordinates, and has a similar Hamiltonian form with the kinetic energy being a sum of squared conjugate momenta. The separable Hamiltonian system was simulated using a high order explicit symplectic integrator.

Different orders of masses yield different sets of Jacobi coordinates. The mass order in this video was selected only based on aesthetic visual presentation, and may contradicts the preferred order found in most literature.

The simulation was performed and rendered in real time.

🎵 "Z-TecH 1" by "Svenzzon" | CC | not affiliated with/endorsed by.

[josepfontanamcnally]

Awesome!! I'm so happy you made another celestial mechanics video!

[Zymplectic]

It should be noted that while stable binary satellite systems may likely exist with mass ratios close to 1, it would be exceptionally rare for triple planets to form or exist. While theoretically stable, small perturbations or slightly different masses may render the system unstable.

[rektlzz]

Nice seeing the N-body problem after a while. I remember asking help from you for the three body problem haha. I figure that you used a 5th order symplectic method I suppose, right ? How accurate is the real-time simulation vs the actual solution that it will take ? When I did similar stuff, the configuration slowly drifted away from the actual solution, the more brutal the gravitational pull between the bodies the faster it diverged from the real solution.

[mahmoudamr]

what is the software you used to do this simulation? thank you