三维梯度系统的双Hamilton结构与势面几何
Bi-Hamiltonian Structure of Gradient Systems in Three Dimensions and Geometry of Potential Surfaces
摘要：
利用Frenet-Serret框架下的双Hamilton结构和雅可比恒等式，证明了三维动力系统具有两个相容的Poisson结构.研究了梯度系统的势函数所定义的曲面的几何与其双Hamilton结构之间的关系。我们证明了在势面上找到梯度流的测地曲率为零的Hamilton函数是可能的。利用这一点，我们得出结论，哈密顿函数是由距离函数的潜在的表面。我们应用这种技术来寻找守恒量的三维梯度系统，包括阿斯特丽德模型的三体运动。
英文摘要：
Working bi-Hamiltonian structure and Jacobi identity in Frenet-Serret frame associated to a dynamical system, we proved that all dynamical systems in three dimensions possess two compatible Poisson structures. We investigate relations between geometry of surfaces defined by potential function of a gradient system and its bi-Hamiltonian structure. We show that it is possible to find Hamiltonian functions whose gradient flows have geodesic curvature zero on potential surfaces. Using this, we conclude that Hamiltonian functions are determined by distance functions on potential surfaces. We apply this technique to find conserved quantities of three dimensional gradient systems including the Aristotelian model of the three-body motion.
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