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Bead pendulum

Simulation of a frictionless bead on 2D and 3D curves mounted to a pendulum. The pendulum bob and the bead with the same mass are affected by gravity. The Hamiltonian of the system with two degrees of freedom was obtained for the general smooth 3D curve in parametric form, which in this demonstration is interpolated between control points (grey spheres). Some video segments feature an interpolated lemniscate. The interpolated curve is based on a spline-like technique acting as third order interpolation with the atomic up(x) function as a mollifier. More precisely, three coefficients for a cubic spline satisfy the quadratic conditions with addition of five consecutive points being used to satisfy a third order condition. An interpolated value between two control points are found as the smooth transition between the two third order polynomials with up(x) as the transition function, with the resulting curve having continuous derivatives of all orders. The canvas displays the system trajectory in terms of the bead and angle coordinate, demonstrating regular and chaotic behavior of the system. The system was simulated using high order explicit symplectic integrators and was rendered in real time. 🎵 "f0LL0w THE LiGHT" by "SofT MANiAC" | CC | not affiliated with/endorsed by.