Integrability of Nonholonomic Heisenberg Type Systems

TL;DR

This paper demonstrates how advanced geometric methods from Hamiltonian dynamics can be applied to analyze nonholonomic Heisenberg type systems, revealing their integrability properties.

Contribution

It introduces the application of geometric techniques such as Killing tensors and Lax pairs to nonholonomic systems derived from Hamiltonian reductions.

Findings

Construction of characteristic Killing tensors for the systems

Identification of compatible Poisson brackets

Development of Lax matrices and classical r-matrices

Abstract

We show that some modern geometric methods of Hamiltonian dynamics can be directly applied to the nonholonomic Heisenberg type systems. As an example we present characteristic Killing tensors, compatible Poisson brackets, Lax matrices and classical $r$-matrices for the conformally Hamiltonian vector fields obtained in a process of reduction of Hamiltonian vector fields by a nonholonomic constraint associated with the Heisenberg system.