Multi-Hamiltonian Structure of Lotka-Volterra and Quantum Volterra Models

TL;DR

This paper explores the Hamiltonian structure of Lotka-Volterra models, revealing conditions for multiple conserved quantities and analyzing a special case linked to the Liouville model in both classical and quantum contexts.

Contribution

It provides a detailed analysis of multi-Hamiltonian formulations of Lotka-Volterra equations and their connection to Liouville models, extending understanding to quantum systems.

Findings

Identification of conditions for multi-Hamiltonian structures

Analysis of a special Liouville-related case

Classical and quantum formulation of the model

Abstract

We consider evolution equations of the Lotka-Volterra type, and elucidate especially their formulation as canonical Hamiltonian systems. The general conditions under which these equations admit several conserved quantities (multi-Hamiltonians) are analysed. A special case, which is related to the Liouville model on a lattice, is considered in detail, both as aclassical and as aquantal system