Some Applications of Affine in Velocities Lagrangians in Two-dimensional Systems

TL;DR

This paper develops a method to construct affine Lagrangians for 2D first-order systems using Jacobi multipliers, with applications in mechanics and biology.

Contribution

It introduces a novel approach to find affine Lagrangians for 2D systems based on Jacobi multipliers, expanding the tools for inverse problems in mechanics and biology.

Findings

Method to construct affine Lagrangians developed

Applications demonstrated with mechanical and biological systems

Examples illustrate the effectiveness of the approach

Abstract

The 2-dimensional inverse problem for first-order systems is analysed and a method to construct an affine Lagrangian for such systems is developed. The determination of such Lagrangians is based on the theory of the Jacobi multiplier for the system of differential equations. We illustrate our analysis with several examples of families of forces that are relevant in mechanics, on one side, and of some relevant biological systems, on the other.