We present a Rayleigh–Ritz variational model of the electron sheath in crossed-field devices. Using laminar layer orbits, this model predicts equilibria in agreement with the classical Brillouin model. Other predictions include the existence of existence of non-classical, stable equilibria as well as limit cycles. This variational model also predicts the existence of a stable propagating diocotron mode, which is absent from the Brillouin model but is present in the double-box model of Kaup and Thomas. Comparisons of these results with those of the cold-fluid theory are discussed.