= Rotor-chain phase-number hydrodynamics
{title2=$H_{\rm harm}=\int dx[\hbar^2(\delta n)^2/(2I)+J(\partial_x\phi)^2/2]$}
With $\delta n=\ell/\hbar$, number and compact phase are conjugate variables. Their long-wavelength <Hamiltonian> has compressibility coefficient $I/\hbar^2$ and <phase stiffness> J. The canonical equations $\dot\phi=\hbar\delta n/I$ and $\dot{\delta n}=J\partial_x^2\phi/\hbar$ combine into $\phi_{tt}=(J/I)\phi_{xx}$. The resulting phase-density wave has physical sound speed $\sqrt{J/I}$ at unit spacing. A <superfluid> has the same low-energy structure even though the rotor coordinate is not the displacement of a mass on a spring.
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