Lagrange-Hamilton geometry applied to a Lotka-Volterra dynamical system
Ana-Maria Boldeanu, Mircea Neagu
TL;DR
This work reframes a three-species Lotka–Volterra system within Lagrange–Hamilton geometry using a least-squares variational approach. It derives a Lagrangian $L$ whose minimizers recover LV trajectories and builds the associated nonlinear connection, Cartan connection, and d-torsions, together with a Lagrangian Yang–Mills–like energy, enabling Jacobi stability analysis via the deviation tensor $P$. It then passes to a Hamiltonian formulation with $H$ and its cotangent-bundle nonlinear connection, obtaining analogous geometric objects and linking them to the antisymmetric part of the Jacobian. The paper closes by identifying constant-level energy surfaces, $\Sigma_{\rho}$, as quadrics and outlining future work to interpret these geometric constructions in the ecological LV context.
Abstract
The aim of this paper is to develop, via the least squares variational method, the Lagrange-Hamilton geometry (in the sense of nonlinear connections, d-torsions and Lagrangian Yang-Mills electromagnetic-like energy) produced by a Lotka-Volterra dynamical system, a simple model of the population dynamics of species competing for some common resource. From a geometrical point of view, the Jacobi stability of this system is discussed.