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 decompo- sition of the bracket into a sum of a Poisson one and an almost Poisson one. The corresponding rela- tion 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’s 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 describe the free motion of some non-self-adjoint systems. The decomposition of the almost Poisson bracket di- rectly 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.
This paper constructs an almost-Poisson structure for the non-self-adjoint dynamical systems, which can be decomposed into a sum of a Poisson bracket and the other almost-Poisson bracket. The necessary and sufficient condition for the decomposition of the almost-Poisson bracket to be two Poisson ones is obtained. As an application, the almost- Poisson structure for generalised Chaplygin＇s systems is discussed in the framework of the decomposition theory. It proves that the almost-Poisson bracket for the systems can be decomposed into the sum of a canonical Poisson bracket and another two noneanonical Poisson brackets in some special cases, which is useful for integrating the equations of motion.
The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of mathematical pendulum shows that the discrete variational method is as effective as symplectic scheme for the autonomous Birkhoff systems.
