The Least Action-Augmented Lanchester Model

Abstract

The principle of least action, a fundamental principle in variational mechanics with broad applicability to classical physical systems, is employed to formulate a novel attrition model for combat dynamics. This formulation extends the Lanchester's square law through second-order temporal derivatives by requiring the resultant Euler-Lagrange equation to coincide with the classical Lanchester's equation. Initial conditions at a specified temporal point enable determination of subsequent system evolution through action minimization, while terminal boundary conditions permit backward reconstruction of combat trajectories. The model's validity is examined through historical analysis of WWII engagements: the Battle of Kursk and the Battle of Iwo Jima. Comparative studies with conventional Lanchester's square models demonstrate marked improvements in predictive accuracy regarding force strength progression, particularly in capturing non-linear attrition patterns characteristic of prolonged engagements.