Decomposition of Almost Poisson Structure of Non-Self-Adjoint Dynamical Systems

TL;DR

This paper develops a decomposition framework for almost Poisson structures in non-self-adjoint dynamical systems, linking them to symplectic and Poisson structures, and applies this to nonholonomic systems for simplified analysis.

Contribution

It introduces a decomposition method for almost Poisson brackets into Poisson and almost Poisson parts, enhancing understanding of their structure and dynamics.

Findings

Decomposition of almost Poisson brackets into Poisson and almost Poisson components.

Application to Chaplygin nonholonomic systems for structure analysis.

Simplification of dynamical vector field integration using decomposition.

Abstract

Non-self-adjoint dynamical systems, e.g., nonholonomic systems, can admit an almost Poisson structure, which is formulated by a kind of Poisson bracket satisfying the usual properties except for the Jacobi identity. A general theory of the almost Poisson structure is investigated based on a decomposition of the bracket into a sum of a Poisson one and an almost Poisson one. The corresponding relation between Poisson structure and symplectic structure is proved, making use of Jacobiizer and symplecticizer. Based on analysis of pseudo-symplectic structure of constraint submanifold of Chaplygin nonholonomic systems, an almost Poisson bracket for the systems is constructed and decomposed into a sum of a canonical Poisson one and an almost Poisson one. Similarly, an almost Poisson structure, which can be decomposed into a sum of canonical one and an almost Lie-Poisson one, is also constructed on an affine space with torsion whose autoparallels are utilized to described the free motion of some non-self-adjoint systems. The decomposition of the almost Poisson bracket directly leads to a decomposition of a dynamical vector field into a sum of usual Hamiltionian vector field and an almost Hamiltonian one, which is useful to simplifying the integration of vector fields.