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 Incoming links: Shift current of an elastic rod

Past exam of the theoretical physics I course of the University of Cambridge / 2017 / 1 / b / Uniform field shift 2026-10-18

The transformation

ψ \mapsto ψ + b

leaves both derivatives unchanged, so the Lagrangian density and action are invariant. For infinitesimal unit field variation, the Noether current for a first-derivative scalar field is the shift current of an elastic rod,

j_{1}^{0} = A ρ \dot{ψ} = Π, j_{1}^{z} = - A K ψ^{'}, Q_{1} = \int_{- \infty}^{\infty} Π d z .

Its divergence vanishes by the equation of motion. The charge is the total longitudinal mechanical momentum to this order in the displacement field.

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