Simulation of a spring with two bobs of equal mass affected by gravity and sliding frictionless on rails. The system system has one degree of freedom for each bob.

The spline-like 2D rail is constructed using 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 for various energies. Tracks of the conjugate momenta are displayed on the canvas revealing either chaotic motion or periodic and regular motion.

0:00 single rail, closed loop, regular
0:22 single rail, modified loop, regular (singular)
0:40 straight and loop rail, chaotic
0:59 different loops, regular
1:15 straight and loop rail, chaotic (continued)
1:21 ellipse rails, chaotic
1:31 identical loops, regular-regular-regular-chaotic

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

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

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