Particle dynamics on a rotating saddle

Particle dynamics on the surface of a rotating saddle simulated with high order symplectic integrators. The system is treated as a Hamiltonian system with a potential x^2 - y^2 rotated around the center. The Hamiltonian is non-autonomous in the Cartesian form because the rotation depends on explicitly on time. The Hamiltonian is generally not constant.

High rotation speed traps the particles with various motions around the center, whereas the particles may consistently roll down or up the surface if the rotation speed is too low.

The saddle potential, in spite of its simplicity, has numerous applications related to particle traps and mass filters albeit with the surface being associated with field potentials rather than height. The saddle shape potential field in particular can arise simply from the presence of four charged rods.

0:00 non-rotating saddle (particles not trapped)
0:07 very slow rotation (particles not trapped)
0:16 rapid rotation
0:22 moderate rotation
0:48 moderate rotation
0:56 slow rotation
1:04 phase space (q1,p2) of slow rotation

[Zymplectic]

PSA: Dislikes being hidden from now on is none of my making. It's a new Youtube thing.
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[LittleJohn]

Im just a curious lay person, so apologies if this is not the right field, but would this also be a good model for lagrange points in astrophysics?

[zoltankurti]

Are you sure you are solvimg the correct equations? The moderate rotation one looks very strange to me. I feel like the balls should start to rotate with the surface and fall down.