The Volterra Integrable case

TL;DR

This paper revisits the integrable Hamiltonian N-species Volterra system introduced by Vito Volterra, analyzing conserved quantities, solutions, and properties of periodic orbits to deepen understanding of its dynamics.

Contribution

It provides a detailed analysis of the integrable case, focusing on conserved quantities and periodic orbit properties, complementing broader research on stationary state models.

Findings

Identification of conserved quantities in the integrable Volterra system

Analysis of period and frequency of periodic orbits

Insights into the solutions of the equations of motion

Abstract

In this short note we reconsider the integrable case of the Hamiltonian N-species Volterra system, as it has been introduced by Vito Volterra in 1937. In the first part, we discuss the corresponding conserved quantities, and comment about the solutions of the equations of motion. In the second part we focus our attention on the properties of the simplest model, in particular on period and frequencies of the periodic orbits. The discussion and the results presented here are to be viewed as a complement to a more general work, devoted to the construction of "a global stationary state model for a sustainable economy in the Hamiltonian formalism".