Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 1 / b / Solution

Stability requires $\lambda >0$. The classical potential $r\rho +\lambda {\rho }^2$ has a symmetry-breaking minimum when $r<0$, atWriting $\rho =\bar{\rho }+\delta \rho$ and discarding constants and total derivatives gives the quadratic LagrangianThe density fluctuation is a gapped amplitude mode, while the phase is the prospective Goldstone boson.