Bi-Hamiltonian structure as a shadow of non-Noether symmetry

TL;DR

This paper explores the link between non-Noether symmetries and bi-Hamiltonian structures, demonstrating that such structures in Hamiltonian systems arise from symmetries of the solution space, exemplified by the KdV equation.

Contribution

It establishes a connection between non-Noether symmetries and bi-Hamiltonian structures, highlighting the role of solution space symmetries in Hamiltonian systems.

Findings

Bi-Hamiltonian structures are caused by symmetries of the solution space.

The paper discusses the bi-Hamiltonian realization of the KdV equation.

Presence of global bi-Hamiltonian structure is linked to non-Noether symmetries.

Abstract

In the present paper correspondence between non-Noether symmetries and bi-Hamiltonian structures is disscussed. We show that in regular Hamiltonian systems presence of the global bi-Hamiltonian structure is caused by symmetry of the space of solution. As an example well known bi-Hamiltonian realisation of Korteweg- De Vries equation is disscussed.