Relations between different Hamiltonian forms of the third Painlev\'e equation

TL;DR

This paper explores various Hamiltonian formulations of the third Painlevé equation, establishing explicit transformations between polynomial and rational Hamiltonian systems to deepen understanding of their interrelations.

Contribution

It introduces explicit birational transformations linking different Hamiltonian forms of the third Painlevé equation, enhancing the comprehension of their structural relationships.

Findings

Explicit birational transformations between Hamiltonian forms

Connections established between polynomial and rational Hamiltonians

Enhanced understanding of the third Painlevé equation's Hamiltonian structures

Abstract

In this paper we study different Hamiltonian systems with polynomial and rational Hamiltonians associated with the generic third Painlev\'e equation and present explicit birational transformations relating them.