Birkhoff系统的广义Hamilton结构与积分不变量

Generalized Hamiltonian structure of Birkhoffian systems and integral invariants

如果不采用对坐标的Darboux变换 ,则动力学系统的Hamilton结构仅适用于保守系统 .通过引入哑变量 ,将Birkhoff系统的动力学空间推广到广义相空间 ,构造出一般系统的Hamilton正则结构 .由于系统拥有辛结构 ,因而系统存在Poincar啨 Cartan积分不变量 .最后 。

Hamiltonian structure of dynamical systems is only suitable to conservative ones if Darboux's transformations are not introduced.Although an extension of Hamiltonian formulation to Birkhoffian one preserves the symbiotic charactor among derivability from a variational principle,Lie algebra and symplectic geometry,without changing the physical meaning of variables and dynamical functions of the systems,the natural sympletic structure as Hamiltonian systems which don't contains dynamical content cannot be realized locally.However,based on an extension of dynamic space of Birkhoffian systems to a generalized phase space by introducing dummy variables,a Hamiltonian canonical structure of general systems is constructed.Since the systems admit symplectic structure,there exist integral invariants of Poincaré and Poincaré cartan's type for the systems.Finally the theory is verified by an illustrative example.