  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.
