Dynamically consistent shallow-atmosphere equations with a complete Coriolis force

Shallow-atmosphere equations retaining both the vertical and horizontal components of the Coriolis force (the latter being neglected in the traditional approximation) are obtained. The derivation invokes Hamilton's principle of least action with an approximate Lagrangian capturing the small increase with height of the solid-body velocity due to planetary rotation. The conservation of energy, angular momentum and Ertel's potential vorticity are ensured in both quasi- and non-hydrostatic systems.