Decomposition of almost Poisson structure of non-self-adjoint dynamical systems  

作　　者：GUO YongXin LIU Chang LIU ShiXing CHANG Peng 

机构地区：[1]College of Physics,Liaoning University,Shenyang 110036,China [2]Department of Applied Mechanics,Beijing Institute of Technology,Beijing 100081,China

出　　处：《Science China(Technological Sciences)》2009年第3期761-770,共10页中国科学（技术科学英文版）

基　　金：Supported by the National Natural Science Foundation of China (Grant Nos. 10872084, 10472040);the Outstanding Young Talents Training Fund of Liaoning Province of China (Grant No. 3040005);the Research Program of Higher Educa-tion of Liaoning Province of China (Grant No. 2008S098)

摘　　要：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.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’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 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.

关 键 词：almost-Poisson structure non-self-adjointness NONHOLONOMIC systems SYMPLECTIC form JACOBI identity torsion 

分 类 号：TB122[理学—工程力学]