论文标题
sasakian结构与二阶Ode和Hamiltonian动力学系统相关
Sasakian structure associated with a second order ODE and Hamiltonian dynamical systems
论文作者
论文摘要
我们在歧管上定义了接触度量结构，对应于二阶的普通微分方程$ d^2y/dx^2 = f（x，y，y，y'）$，并显示触点度量结构是Sasakian，并且仅当1型$ \ frac $ \ frac {1} {1} {2} {2}（dp-fdx）$定义一个Poisson结构。我们考虑了由这种泊松结构定义的汉密尔顿动力系统，并表明，哈密顿矢量场是Reeb矢量场的倍数，它承认可以将$ F $视为汉密尔顿功能的兼容Bi-Hamiltonian结构。作为一种特殊情况，我们给出了Reeb矢量场的兼容双汉米尔顿结构，使得结构方程与Heisenberg群体上不变的Coframe的Maurer-Cartan方程相对应，并且自变量可在Hamiltonian功能中起作用。我们还表明，对于某些$ψ$，如果$ f_x+ff_p =ψ（x）$，reeb vector场的整体曲线的正常束的第一类消失了。
We define a contact metric structure on the manifold corresponding to a second order ordinary differential equation $d^2y/dx^2=f(x,y,y')$ and show that the contact metric structure is Sasakian if and only if the 1-form $\frac{1}{2}(dp-fdx)$ defines a Poisson structure. We consider a Hamiltonian dynamical system defined by this Poisson structure and show that the Hamiltonian vector field, which is a multiple of the Reeb vector field, admits a compatible bi-Hamiltonian structure for which $f$ can be regarded as a Hamiltonian function. As a particular case, we give a compatible bi-Hamiltonian structure of the Reeb vector field such that the structure equations correspond to the Maurer-Cartan equations of an invariant coframe on the Heisenberg group and the independent variable plays the role of a Hamiltonian function. We also show that the first Chern class of the normal bundle of an integral curve of a multiple of the Reeb vector field vanishes iff $f_x+ff_p = Ψ(x)$ for some $Ψ$.
