Poissonization of Three Dimensional Nonholonomic Dynamics with the Method of Extension

TL;DR

This paper introduces a systematic method to construct Poisson operators for three-dimensional nonholonomic systems by embedding them into larger spaces, enabling Poissonization without relying on the Hamiltonian's specific form.

Contribution

It presents a novel extension-based approach to Poissonization of nonholonomic dynamics, avoiding symmetry reduction and ensuring the Jacobi identity through conformal factors.

Findings

Successfully constructs Poisson operators for nonholonomic systems.

Enables calculation of equilibrium distributions in non-Hamiltonian ensembles.

Provides a general framework independent of Hamiltonian form.

Abstract

In this study we develop a systematic procedure to construct a Poisson operator that describes the dynamics of a three dimensional nonholonomic system. Instead of reducing by symmetry the antisymmetric operator that links the energy gradient to the velocity on the tangent bundle, the system is embedded in a larger space. Here, the extended antisymmetric operator, which preserves the original equations of motion, satisfies the Jacobi identity in a conformal fashion. Thus, a Poisson operator can be obtained by a further time reparametrization. Such Poissonization does not rely on the specific form of the Hamiltonian function. The theory is applied to calculate the equilibrium distribution function of a non-Hamiltonian ensemble.