Hamiltonian Systems On Symplectic Vector Spaces

The symplectic vector space is a kind of vector space with the symplectic structure.The symplectic structure is skew symmetric coupling structure,which is the first studied by Abel.Because the skew symmetric structure is confused with the name of complex structure,Weyl named the skew symmetric coupling structure as symplectic structure in 1938.The Chinese name of symplectic structure is translated according to the pronunciation of symplectic by Luogeng Hua in 1944.In fact,the phase space of analytical mechanics is exact the symplectic space.But most of the great masters of classical mechanics pay more attention to the analytic formulations not the global formulation of dynamical systems.And lots of very useful analytic methods are be given which can be used to study the local characteristic of symplectic structure.Hamiltonian mechanics is a kind of classical theory on phase space of dynamical systems.The application fields of Hamiltonian mechanics are expanded because the geometric structure of Hamiltonian mechanics is symplectic structure.Such as Hamiltonian mechanics is used not only in classical physics but also in modern physics,not only in macroscopic fields but also in microscopic fields and cosmoscopic fields,not only in deterministic problems but also in stochastic problems,not only in continuous problems but also in discrete problems,not only in holonomically constrianed problems but also in nonholonomically constrained problems.In this paper,the symplectic geometric structure and Poisson geometric structure in vector space are studied,as well as the geometric structure of symplectic vector space and Poisson vector space and its subspace.Based on the geometric structure of symplectic vector space and Poisson vector space,the geometric structure and dynamics of linear Hamilton system are studied.The research contents are following :Firstly,the geometric structure of symplectic vector space,its subspace theory,and the reduction problem are studied.Secondly,geometric structure of Poisson vector space,its subspace theory,and the reduction problem are studied.Finally,the linear Hamilton mechanics of the symplectic vector space and Poisson vector space is studied.The Lie algebraic structure of the smooth function and Hamiltonian vector field is also studied.The canonical transform theory of the Hamilton system is discussed in detail.