The restoring torque on a heavily saturating compass needle is modelled

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Nonlinear dynamics and chaos

Error, Chance and the Machine, Ch Iteration: nonlinear systems and eigenvalues

Problem

The restoring torque on a heavily saturating compass needle is modelled, in dimensionless units, by τ(θ)=tanh⁡θ\tau(\theta) = \tanh\theta, with equilibrium where τ(θ)=0\tau(\theta)=0. Someone applies Newton's method from θ0=2\theta_0=2: θ1=θ0−tanh⁡θ01−tanh⁡2θ0=2−0.96400.07065=−11.64\theta_1 = \theta_0 - \dfrac{\tanh\theta_0}{1-\tanh^2\theta_0} = 2 - \dfrac{0.9640}{0.07065} = -11.64, and the next step jumps to θ2≈3.3×109\theta_2 \approx 3.3\times10^{9}. They conclude: "the iterates run away to infinity, so τ(θ)=0\tau(\theta)=0 has no solution." ((a)) State the actual equilibrium angle θ∗\theta^* and explain why it exists despite the observed divergence. ((b)) Explain, using the sufficient-decrease idea of a damped Newton step, why the very first step above would be rejected, and why damping (rather than the conclusion of no root) is the appropriate remedy.

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