Geodesic sprays and frozen metrics in rheonomic Lagrange manifolds

TL;DR

This paper introduces a framework for modeling wildfire spread using Hamilton orthogonal nets derived from rheonomic Lagrange manifolds, and shows how to obtain associated frozen Finsler metrics that simplify analysis.

Contribution

It defines energy pre-extremals and Hamilton orthogonal nets in rheonomic Lagrange manifolds, and introduces the concept of frozen metrics for simplifying time-dependent problems.

Findings

Energy pre-extremals become Finsler geodesics in frozen metrics.

Hamilton orthogonality is preserved during the freezing process.

Analytic and numerical solutions for 2D Randers spaces are provided.

Abstract

We define systems of pre-extremals for the energy functional of regular rheonomic Lagrange manifolds and show how they induce well-defined Hamilton orthogonal nets. Such nets have applications in the modelling of e.g. wildfire spread under time- and space-dependent conditions. The time function inherited from such a Hamilton net induces in turn a time-independent Finsler metric - we call it the associated frozen metric. It is simply obtained by inserting the time function from the net into the given Lagrangean. The energy pre-extremals then become ordinary Finsler geodesics of the frozen metric and the Hamilton orthogonality property is preserved during the freeze. We compare our results with previous findings of G. W. Richards concerning his application of Huyghens' principle to establish the PDE system for Hamilton orthogonal nets in 2D Randers spaces and also concerning his explicit spray solutions for time-only dependent Randers spaces. We analyze examples of time-dependent 2D Randers spaces with simple, yet non-trivial, Zermelo data; we obtain analytic and numerical solutions to their respective energy pre-extremal equations; and we display details of the resulting (frozen) Hamilton orthogonal nets.