The last four equations describe the evolution of slowly varying wave trains over bathymetry in interaction with the mean flow, and can be derived from a variational principle: Whitham's average Lagrangian method.
Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing: Saarbrücken, 2013.
Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface.
Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.
The most commonly used tools are Lagrangians and Hamiltonians and are used to derive the Einstein field equations.