Lagrangian formulation, a general relativity analogue, and a symmetry of the Vialov equation of glaciology

TL;DR

This paper establishes a Lagrangian framework for the Vialov equation in glaciology, revealing a formal analogy with cosmological equations and uncovering a new symmetry that enables solution generation.

Contribution

It introduces a Lagrangian formulation for the Vialov equation, linking it to cosmological models and identifying a novel symmetry for solution derivation.

Findings

Found a Lagrangian for the Vialov equation.

Established an analogy with the Friedmann equation.

Discovered a new symmetry allowing solution generation.

Abstract

Using a suitable rescaling of the independent variable, a Lagrangian is found for the nonlinear Vialov equation ruling the longitudinal profiles of glaciers and ice caps in the shallow ice approximation. This leads to a formal analogy between the (rescaled) Vialov equation and the Friedmann equation of relativistic cosmology, which is explored. This context provides a new symmetry of the (rescaled) Vialov equation and gives, at least formally, all its solutions using a generating function, which is the Nye profile for the degenerate case of perfectly plastic ice.