Mechanics of Helix Under Gravity

This lecture focuses on the mechanics of a helix under gravity, studying the equilibrium shapes of helical thin rods using Kirchhoff's rod theory. The analysis includes the evolution of the Cosserat frame, the equilibrium shape determination, and the constitutive relations for internal moments. The lecture delves into rescaling quantities, computing derivatives of the internal moment vector, and solving ordinary differential equations for the helix components. It also covers the weak gravity limit, perturbative solutions, and the computation of the helix shape under gravity. The lecture concludes with the integration of differential equations to determine the displacement of the free end due to gravity.