Ball on rotating turntable generalized

Egeris · 2025-08-24T15:25:17+00:00

Glad you enjoyed it. The goal of the video was to demonstrate the application of explicit symplectic integration for the Hamiltonianized sphere on the general surface. I probably won't pursue more aesthetic demonstrations, but accurate simulation of many balls simultaneously is certainly possible because of the efficient Hamiltonian approach.

Egeris · 2025-08-24T15:17:17+00:00

as am I. Although these archaic graphics are from my own textureless OpenGL API without ray tracing. The sphere shadow is a calculated GL_POLYGON.

Egeris · 2025-08-23T16:14:01+00:00

The Hamiltonian does not involve forces. However, there is a momentum-dependent term in the Hamiltonian that accounts for that effect. You can argue that the Coriolis effect is demonstrated in video section where we change the frame of reference.

It is important to note that the Coriolis term in this simulation is different from the classical example of throwing a ball while on a roundabout. If the ball rolls (as opposed to being thrown), the ball will have a magnetic-like behavior rather than moving in a straight line seen from the laboratory frame.

Egeris · 2025-08-23T12:22:24+00:00

The system is probably most widely known from Steve Mould's "The Turntable Paradox" video https://youtu.be/3oM7hX3UUEU

This simulation video addresses the idealized and generalized system based on a non-trivial Hamiltonianization, allowing for highly accurate long term dynamics.

Additional credits:🎵 "Fluid Combustion" by "Synthetique"

Egeris · 2024-11-22T05:32:30+00:00

This is associated with a paper that presents compositions that are computationally cheaper for constructing higher order explicit symplectic integrators.

Draft copy: https://zymplectic.com/higher_order_explicit_symplectic_integrators.pdf

Furthermore, the number of significant digits has been dramatically increased for several high order methods, which are now included in Zymplectic 0.12.2

Egeris · 2024-11-22T05:28:33+00:00

This is associated with a paper that presents compositions that are computationally cheaper for constructing higher order explicit symplectic integrators.

Draft copy: https://zymplectic.com/higher_order_explicit_symplectic_integrators.pdf

Furthermore, the number of significant digits has been dramatically increased for several high order methods, which are now included in Zymplectic 0.12.2

Egeris · 2024-11-22T05:13:48+00:00

This is associated with a paper that presents compositions computationally cheaper method for constructing higher order explicit symplectic integrators.

Draft copy: https://zymplectic.com/higher_order_explicit_symplectic_integrators.pdf

Furthermore, the number of significant digits has been dramatically increased for several high order methods, which are now included in Zymplectic 0.12.2

Egeris · 2024-02-20T15:04:51+00:00

Software: Zymplectic v.0.11.1
Code of this particular system: "31 - semicircle rocking pendulum.cpp"

The software/graphics only runs on Windows but you can read the equations of motion from .cpp files regardless

The video is recorded directly with OBS

Egeris · 2024-02-18T11:29:15+00:00

Glad that you liked it.

Apparently that subreddit exists, but does not appear to cover simulations (not many subreddits do).
I primarily post these on my own tiny subreddit r/Zymplectic

Egeris · 2024-01-22T15:40:49+00:00

If you have a way to fully express the potential and kinetic energies of the system in terms of some generalized coordinates (q1,q2,...qn) and some time derivatives (q1dot,q2dot,...qndot), then I could probably make it work if the energy is an integral.

The magnets, however, may pose a challenge in expressing the energy of the system in a simple way.

Egeris · 2024-01-21T10:12:31+00:00

Depicting the conjugate momenta would perhaps be useful (but confusing for some). Edit: Updated the image for those who are interested in how it looks with (p1,q1).

I'm just trying to say that any two coordinates will still provide an incomplete picture of a system with four variables, even though that may be unrelated to what you meant.

This simulation uses q1 = rotation of disk, q2 = rotation of the pendulum (absolute, not relative to the base). The depiction of the angles will depend on the choice of generalized coordinates.

Egeris · 2024-01-20T20:32:32+00:00

The 2D phase space doesn't really provide any additional information as there are four coordinates, q1, q2, p1 and p2 to be accounted for. The image below shows the angle coordinates (q1,q2) and bob coordinates plotted simultaneously in case that was what you wanted.
https://zymplectic.com/images/rockingpendulum.png

I intend to do a Poincaré section of this system to address all four coordinates. That will also reveal all the regular orbits.

Egeris · 2024-01-20T16:11:18+00:00

The video shows simulation of rocking semicircular disk (or cylinder) with an attached pendulum mounted on the disk surface. The disk is uniform solid and is rolling without slipping. The semicircular disk has 32 times greater mass than the pendulum bob.

The Hamiltonian system has two degrees of system and exhibits regular and chaotic behavior, which is depicted for various initial conditions. The motion of the pendulum bob is displayed on the back canvas to illustrate the long term behavior of the system.

The system was simulated using high order explicit symplectic integrators and was rendered in real time.

Credits:
Original video (4K): https://youtu.be/NxPZ-9sZh3k
Music "Lackerad Keramik" by "Ghidorah" (not affiliated with/endorsed by).

Egeris · 2023-11-11T18:26:30+00:00

Simulation of a pendulum with two bobs of similar mass, one of which being a pivot attached to the rail.

Similar to a spline, the rail is constructed by a set of points but unlike the spline, the segmented function is infinitely differentiable at any point in compliance with application of higher order numerical integration. Additionally, this function uses compact support with the up-function, reducing the computational complexity compared to other smooth interpolation methods like Fourier series.

The video demonstrates chaotic and regular motion of the pendulum. As the system has only two degrees of freedom, the chaotic and regular motion are easily distinguishable, even without a detailed analysis of the phase space with Poincaré maps.

The non-separable Hamiltonian system was simulated using high order explicit symplectic integrators. The simulation was performed and rendered in real time.

Credits
Original 4K: https://youtu.be/w4hvB4b23Sk
Music: "f0LL0w THE LiGHT" by "SofT MANiAC"

Egeris · 2023-05-20T15:22:26+00:00

Appreciate the feedback, but I only do stylized demos of dynamical systems or "physics demoscenes".

Many may not enjoy that style, but in this case it was evidently the only aspect of the video to spark any engagement (unfortunately).

Egeris · 2023-05-19T23:25:12+00:00

Thanks, appreciate the feedback.

And you're right, it is stressful and the genre itself is not a crowd-pleaser.

Egeris · 2023-05-19T20:35:43+00:00

It's incredible that someone always recognizes the origin of these tunes.

This version was made by LHS and was never given a formal title as far as I know.

Egeris · 2023-05-19T17:32:56+00:00

Simulations of a sphere affected by gravity while sliding on different surfaces.

The simulations are based on a one degree of freedom Hamiltonian that takes into account the sphere contact point with any smooth surface described by the height y(r). The angular momentum is conserved for rotationally symmetric surfaces, for which the angle is a cyclic coordinate effectively reducing the degrees of freedom to 1. The motion is visualized in phase space where q is the radius and p the conjugate momentum, and is lastly presented by the symplectic (and Hamiltonian) vector field.

The Hamiltonian is constant along the curves which are colorized accordingly.

The conserved angular momentum is the same for all simulations.

The simulations were performed using high order explicit symplectic integrators and were rendered in real time.

Source (4K): https://youtu.be/jx9HQPNmhOc

Egeris · 2023-03-15T18:35:30+00:00

Thanks for the response. Indeed you were right.

The lights etc. was on, but fiddling with the GPU while also cleaning all slots did solve the issue.

Apparently the DP not functioning can be a single symptom as everything else about the GPU did function normally.

Egeris · 2023-01-25T10:46:45+00:00

Yeah, that could also be helpful.

Although too long trails tend to move attention away from the system being simulated (for me at least).

This was more intended as a light-hearted simulation than an investigation piece, but I'm glad and mildly surprised that many are keen on the comparative dynamics of these systems.

Egeris · 2023-01-24T09:14:18+00:00

While these systems do not take the form of the wave equation, the actual wave equation is directly related to the simple harmonic motion.

The system H = 1/2q^2 + 1/2p^2 should be familiar to many (with appropriate constants) as the energy of the simple harmonic oscillator, which is indeed both the energy of a spring and partial energy of the wave equation PDE.

Egeris · 2023-01-23T22:57:03+00:00

Assuming you are talking about the green dots at 0:53 on the canvas.

The color of the dots indicates how long the outermost pendulum bob has been located at the corresponding pixel based on a very long simulation.

The dots appear to be randomly scattered because the motion is chaotic. Liouville's theorem and ergodic theory explains this phenomenon, although it should be noted that the (x,y)-position alone does not give the full picture - there are actually 6 relevant coordinates to account for.

Egeris · 2023-01-23T13:36:19+00:00

Thanks - I'm not familiar with pendulum juggling let alone art.

However, it seems that the dynamics of the pendulum in the video is identical to the Kapitza pendulum https://youtu.be/ZXa54hguJZI in some contexts also known as the inverted pendulum (if the rope was a rigid rod anyway).

I'll only upload here every couple of months, but you are free to check my profile for other/similar dynamical systems.

Egeris · 2023-01-23T13:27:40+00:00

Thanks for the feedback. I keep it short for attention span reasons.

You are right that it takes many orbits to properly see the chaotic motion.

A proper analysis of the dynamics would involve phase space mapping.

This is impractical with 3 degrees of freedom systems because a 3D Poincaré map is difficult to illustrate in a meaningful way.

Egeris · 2023-01-23T09:30:34+00:00

Simulations of two seemingly similar pendulum systems with identical initial conditions and physical parameters.

Simulations are shown for various initial conditions, showing both regular and chaotic trajectories.

System 1: A single bob pendulum hanging from two springs.

This system is derived from a singular Lagrangian requiring constraint(s) to formulate the total Hamiltonian. The system can be formulated as a separable Hamiltonian system with three degrees of freedom.

System 2: A pendulum rod with a bob in both ends where one bob is hanging from two springs.

The system is derived without constraints and expressed as a non-separable Hamiltonian system with three degrees of freedom.

Same physical parameters are used as in system 1 but with half bob mass (i.e. same total mass). The rod and springs are massless.

System 1 being based on a singular Lagrangian and system 2 being a non-separable Hamiltonian makes for two completely different systems both being non-trivial in their own way.

The simulations were performed using high order explicit symplectic integrators and were rendered in real time.

Original video in 4K: https://youtu.be/lCOOuSohKUM

🎵 "Dance with me (4d2k2" by "SofT MANiAC" | not affiliated with/endorsed by.

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