An Update on the Lagrangian of the Lotka-Volterra System

1. We present an attempt at finding a Lagrangian for the canonical, two-dimensional Lotka-Volterra (LV) system. In finding its Lagrangian, we first review previous literature on the system's Hamiltonian representation, and then we backcompute its corresponding Lagrangian. We examine the rich mechanical intepretation of the system that the Lagrangian offers. We also discover interesting subtleties in the procedure: It is revealed that the equations of motion for the predator population obtained through the Hamiltonian form and Lagrangian are as one expects, but there is a loss of a key model parameter in the Lagrangian formulation. Taken at face-value, it is found that the mechanics of a "predator particle" correspond to a particle in an exponentially increasing well that is damped or "revved" depending on its position in that well. We then discuss the difficulties that arise when finding a Lagrangian representation of the original LV system. Finally, we discuss an unaccounted-for freedom in the coordinate transformation that leads to a family of Lagrangians that are not trivially equivalent.