Mechanics of a helix under gravity

This lecture explores the equilibrium shapes of helical thin rods under weak gravity, focusing on the computation of their shapes using Kirchhoff's rod theory. The analysis includes the unique representation of the rod's geometry, rescaled Kirchhoff equations, and the derivation of ordinary differential equations. The lecture also covers the solution verification, kinematic equations, and the integration of geometric parameters. Additionally, it delves into the computation of the radius and pitch of the three-dimensional curve, considering the impact of weak gravity conditions on the rod's shape.