On a geometric description of time dependent singular Lagrangians with applications to biological systems

TL;DR

This paper explores a geometric approach to time-dependent singular Lagrangians in biological models, demonstrating how to construct Lagrangians and Hamiltonian descriptions using Jacobi Last Multipliers and Dirac brackets.

Contribution

It introduces a systematic geometric framework for analyzing models with singular Lagrangians, including explicit methods for biological competition and predation models.

Findings

Existence of Jacobi Last Multiplier for the models

Construction of singular Lagrangians from the multiplier

Hamiltonian formulation via Dirac brackets for singular systems

Abstract

We consider certain analytical features of a stochastic model that can explain among other things competition among species and simultaneous predation on the competing species from a geometric perspective which allows for a systematic description of models admitting singular Lagrangians. The model equations are shown to admit a Jacobi Last Multiplier which in turn allows for the construction of a Lagrangian. The Lagrangian is of singular nature so that construction of the Hamiltonian via a Legendre transformation is not possible. A Hamiltonian description of the model therefore requires the introduction of Dirac brackets. Explicit results are presented for the "Kill the winner" model and its reductions.