Analogues of glacial valley profiles in particle mechanics and in cosmology

An ordinary differential equation describing the transverse profiles of U-shaped glacial valleys, derived with a variational principle, has two formal analogies which we analyze. First, an analogy with point particle mechanics completes the description of the solutions. Second, an analogy with the Friedmann equation of relativistic cosmology shows that the analogue of a glacial valley profile is a universe with a future singularity but respecting the weak energy condition. The equation unveils also a Big Freeze singularity, which was not observed before for positive curvature index.