Do Mature, Fluvial Landscapes Obey Hamilton's Principle?

Abstract

Students of physics typically take a theory course on classical mechanics in which they learn about Hamilton's Principle and how it can be used to derive many well-known physical laws that describe the motion of objects from particles, to light rays, to celestial bodies, including Newton's laws and Snell's Law from geometrical optics.  This powerful principle has also been shown to apply to fields (i.e., continuous systems) such as the electromagnetic and gravitational fields, and it is a foundational concept in quantum physics.  Hamilton's principle states that the dynamics of a physical system will optimize a functional (in our case, an integral over a spatial domain) of the system's Lagrangian, which is typically the difference between its kinetic and potential energies.  Many previous authors have postulated that fluvial landscapes may evolve in such a way that local and/or global kinetic energy dissipation or stream power is minimized, and this is the basis of the optimal channel network (OCN) simulation models that have been widely studied.  However, Hamilton's Principle suggests that these formulations are lacking an important piece, namely the global introduction of potential energy into the fluvial system by rainfall.  The author will show that by introducing this missing piece, Hamilton's Principle and the Euler-Lagrange theorem lead to a partial differential equation (PDE) for idealized, steady-state landforms.  This same PDE can also be derived from conservation of mass and an empirical slope-discharge formula.  These connections therefore point to a new theoretical framework for understanding the interplay between function and form in mature, fluvial landforms;  that is, an explanation for why these landforms take the forms we observe.  The author will also present ideas and algorithms for analyzing digital elevation models (DEMs), in an effort to test for agreement with Hamilton's principle.