Poissonization of Three Dimensional Nonholonomic Dynamics with the Method of Extension

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.