Implementing the Equations of Motion in the Energy Line Principle to Simulate the Runout Zones of Gravitational Natural Hazards

Abstract. The mitigation of the risk from gravitational natural hazards involves a variety of measures in engineering and management. The basis for any measures is the identification of hazard prone areas by considering past events and model simulations. Various models are available for the simulation of gravitational natural hazards but their application is often a trade off between simplicity and accuracy. A physical interpretation of the well established energy line principle allows to formulate the corresponding equations of motion and to introduce a model that is still simple but more accurate. The equations of motion are derived by the application of the Lagrange formalism for a friction block that slides on an inclined plane. Furthermore, a numerical algorithm based on the Euler method is set up to solve the equations of motion on a digital terrain model. This model is applied and compared to the energy line principle based on the equation of energy and to past events in two case studies for rockfall and landslide. The outcomes show that the formulation of the energy line principle with the equations of motion corresponds well to the formulation based on the equation of energy but allows a more differentiated simulation of the runout zone that reproduces better the past events. However, there are artifacts from the numerical solution of the equations of motion that require a deeper theoretical investigation and also uncertainties in the past events that must be worked through in a more elaborated empirical study. Nevertheless, the concept of this approach to enhance the performance of the energy line principle simply by another perspective to the same physical concept is thus proved.