Non-Noether symmetries and their influence on phase space geometry

TL;DR

This paper explores the geometric properties of non-Noether symmetries in Hamiltonian systems, revealing their connection to Lax pairs, conservation laws, and involution conditions, thus enriching the understanding of phase space geometry.

Contribution

It establishes a link between non-Noether symmetries and various geometric methods for generating conservation laws, including Lax pairs and bi-Hamiltonian structures, and proves involution of associated integrals.

Findings

Non-Noether symmetries lead to Lax pairs in regular Hamiltonian systems.

Conservation laws from non-Noether symmetries are in involution under certain conditions.

The work connects non-Noether symmetries with established geometric frameworks like bi-Hamiltonian formalism.

Abstract

We disscuss some geometric aspects of the concept of non-Noether symmetry. It is shown that in regular Hamiltonian systems such a symmetry canonically leads to a Lax pair on the algebra of linear operators on cotangent bundle over the phase space. Correspondence between the non-Noether symmetries and other wide spread geometric methods of generating conservation laws such as bi-Hamiltonian formalism, bidifferential calculi and Frolicher-Nijenhuis geometry is considered. It is proved that the integrals of motion associated with the continuous non-Noether symmetry are in involution whenever the generator of the symmetry satisfies a certain Yang-Baxter type equation.