🎯 3D Spherical Pendulum

A spherical pendulum is a bob attached to a fixed pivot by a rigid rod or string of length L, free to swing in any direction rather than being confined to a single vertical plane like a simple pendulum. Mathematically the bob's position is constrained to lie on a sphere of radius L; this simulator enforces that constraint numerically every step by rescaling the RK4-integrated position back onto the sphere and removing the outward-pointing component of velocity.

After each RK4 step the raw position (x,y,z) is rescaled by L divided by its current distance from the origin, snapping it back exactly onto the unit sphere, and the velocity's radial component v·r̂/L is subtracted off so only the tangential part remains. This projection method is simpler than solving a Lagrange-multiplier constraint force analytically and keeps floating-point drift from ever visibly stretching the string.

For a bob to trace a perfect circle at constant polar angle θ, the net inward force from gravity and tension must equal the required centripetal force m·v_φ²/(L·sinθ). Solving for the azimuthal speed gives v_φ = √(gL⋅sinθ⋅tanθ), which is exactly the initial velocity this simulator assigns when you switch to Conical mode — any other azimuthal speed at that angle would make the bob spiral toward a different θ instead of holding steady.

In the pendulum's local (non-inertial) reference frame, the rotating Earth introduces a fictitious Coriolis force 2·Ω×v acting sideways on the moving bob. Only the component of Earth's rotation vector along the local vertical matters for this horizontal-plane drift, and that component has magnitude Ω·sin(latitude) — full strength at the poles, zero at the equator — which this simulator implements directly in its equations of motion via the Omz term multiplying the velocity.

Energy E = ½mv² + mg(z+L) is the bob's kinetic plus gravitational potential energy (zero at the lowest point of the sphere); angular momentum L_z = m(x·v_y − y·v_x) is its rotation about the vertical axis. For planar and conical modes both quantities should stay essentially constant over time, since gravity and the sphere's tension force do no net work and exert no torque about the vertical axis — watching them stay flat is a direct check that the RK4 integrator and sphere-projection constraint are behaving correctly.