Past exam of the theoretical physics I course of the University of Cambridge / 2022 / 4 / d / Solution

The velocity coupling changes the momenta and the undifferentiated field terms. Variation with respect to

ϕ_{2}

gives the equation for

ϕ_{1}

:

\frac{\partial L}{\partial ϕ _{2}} = - \frac{ϕ _{1, t}}{2 τ}, \frac{\partial L}{\partial ϕ _{2, t}} = ϕ_{1, t} + \frac{ϕ _{1}}{2 τ},

so the two Euler-Lagrange field equations are

ϕ_{1, tt} - c^{2} ϕ_{1, xx} + \frac{1}{τ} ϕ_{1, t} = 0, ϕ_{2, tt} - c^{2} ϕ_{2, xx} - \frac{1}{τ} ϕ_{2, t} = 0.

They describe damped and antidamped fields from an antisymmetric velocity coupling. With the convention

ϕ_{j} = A_{j} e^{i (k_{j} x - ω_{j} t)}

, their characteristic equations are

ω_{1}^{2} + \frac{i}{τ} ω_{1} - c^{2} k_{1}^{2} = 0, ω_{2}^{2} - \frac{i}{τ} ω_{2} - c^{2} k_{2}^{2} = 0.

Thus the dispersion relations and modes are

ω_{1, \pm} = - \frac{i}{2 τ} \pm c^{2} k_{1}^{2} - \frac{1}{4 τ ^{2}}, ϕ_{1, \pm} = A_{1, \pm} e^{i (k_{1} x - ω_{1, \pm} t)},

ω_{2, \pm} = + \frac{i}{2 τ} \pm c^{2} k_{2}^{2} - \frac{1}{4 τ ^{2}}, ϕ_{2, \pm} = A_{2, \pm} e^{i (k_{2} x - ω_{2, \pm} t)} .

For each spatial Fourier mode, the general solution is the linear combination of its two temporal branches; spatial superposition gives the general field. Take real parts or suitable complex conjugate pairs to obtain the required real fields. At the repeated-root threshold, the independent second temporal solution contains an extra factor

t

, as detailed next.

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