Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 2 / d

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- Solution d

Solution

Put

q = δ θ

,

p = δ ϕ

, so

θ = π /2 - q

and

ϕ = p

. The quadratic Lagrangian is obtained by taking

L_{2} = s q \overset{p}{˙} - \frac{ρ _{s}}{2} [(\nabla q)^{2} + (\nabla p)^{2}] .

The two real fluctuations form a coordinate and its canonical momentum rather than two independent modes. Their Euler-Lagrange equations combine to give

\overset{q}{¨} + (\frac{ρ _{s}}{s})^{2} \nabla^{4} q = 0,

and similarly for

p

. The ferromagnetic magnon therefore has the quadratic dispersion relation

ω_{k} = \frac{ρ _{s}}{s} k^{2} .

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