The configuration space of a double pendulum is the torus with angle coordinates

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Topology and global structure

Invariance, Ch Manifolds, charts and tangent spaces

Problem

The configuration space of a double pendulum is the torus T2T^2 with angle coordinates (θ1,θ2)(\theta_1,\theta_2), each identified modulo 2π2\pi. Consider the one-form ω=4 dθ1−3 dθ2\omega = 4\,\mathrm{d}\theta_1 - 3\,\mathrm{d}\theta_2 on T2T^2 (its components are constants, so dω=0\mathrm{d}\omega = 0 by the same computation used for constant-coefficient forms in flat space). Let CC be the closed loop θ1(t)=4πt\theta_1(t) = 4\pi t, θ2(t)=2πt\theta_2(t) = 2\pi t for t∈[0,1]t\in[0,1] (so as tt runs from 00 to 11, θ1\theta_1 winds twice around its circle and θ2\theta_2 winds once around its circle, and the loop closes because both angles return to their starting values modulo 2π2\pi). ((a)) Explain, by analogy with the closed but not exact one-form dϕ\mathrm{d}\phi on a punctured plane, why dω=0\mathrm{d}\omega=0 does not by itself guarantee that ∮Cω\oint_C\omega vanishes. ((b)) Evaluate ∮Cω\oint_C \omega by evaluating ω\omega on the tangent vector γ˙(t)=θ˙1 ∂θ1+θ˙2 ∂θ2\dot\gamma(t) = \dot\theta_1\,\partial_{\theta_1} + \dot\theta_2\,\partial_{\theta_2} of the loop and integrating over tt.

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