Involutive orbits of non-Noether symmetry groups

TL;DR

This paper explores the structure of involutive function families generated by non-Noether symmetries on Poisson manifolds, linking these symmetries to integrability in Hamiltonian systems through examples like the Boussinesq and Broer-Kaup systems.

Contribution

It introduces a new class of vector fields related to non-Noether symmetries that generate involutive families of functions on Poisson manifolds, expanding understanding of symmetries in Hamiltonian dynamics.

Findings

Identifies vector fields producing involutive functions on Poisson manifolds.

Establishes relationship between non-Noether symmetries and involutive function families.

Demonstrates concepts with models like the modified Boussinesq and Broer-Kaup systems.

Abstract

We consider set of functions on Poisson manifold related by continues one-parameter group of transformations. Class of vector fields that produce involutive families of functions is investigated and relationship between these vector fields and non-Noether symmetries of Hamiltonian dynamical systems is outlined. Theory is illustrated with sample models: modified Boussinesq system and Broer-Kaup system.