River cross-sections obey the same equation as an expanding universe

In the shallow-water approximation, the equation describing a river's transverse depth profile is mathematically identical to the Friedmann equation for an Anti-de Sitter universe. This formal equivalence yields an unexpected Lagrangian for the river profile, and extremizing the corresponding action turns out to be equivalent to maximizing the friction force on the riverbed and the rate of energy dissipation. A second-variation analysis confirms this extremum is a maximum, not a minimum — meaning rivers settle into the shape that dissipates energy as rapidly as possible.