耗散Hamilton-Poisson系统的渐近稳定性

轴对称力场和陀螺力矩作用下刚体转动运动的Hamilton-Poisson公式

Hamilton-Poisson formulation for the rotational motion of a rigid body in the presence of an axisymmetric force field and a gyroscopic torque

本文给出了在轴对称力场和陀螺力矩作用下，刚体可采用Hamilton-Poisson方程的充分条件。即使通过增加一个陀螺力矩，我们最初失去了一个保守的卡西米尔，我们恢复另一个守恒律作为修改后的泊松结构的卡西米尔函数。我们应用这个框架在文献中的几个众所周知的结果。

关于广义 Poisson 流形上广义协变 Hamilton 系统的研究

A Study of Generalized Covariant Hamilton Systems On Generalized Poisson manifold

针对广义 Hamilton 系统理论框架不完善的问题，通过定义广义梯度算子、广义结构 Poisson 括号和几何括号，推导出广义协变 Hamilton 系统，完善了其基本理论框架。

On the dynamics of a Hamilton-Poisson system

The dynamics of a three-dimensional Hamilton-Poisson system is closely related to its constants of motion, the energy or Hamiltonian function $H$ and a Casimir $C$ of the corresponding Lie algebra. The orbits of the system are included in the intersection of the level sets $H=constant$ and $C=constant$. Furthermore, for some three-dimensional Hamilton-Poisson systems, connections between the associated energy-Casimir mapping $(H,C)$ and some of their dynamic properties were reported. In order to detect new connections, we construct a Hamilton-Poisson system using two smooth functions as its co…

哈密顿泊松积分器的一些稳定性性质

Some stability properties of Hamiltonian Poisson integrators

本文阐述可积与不可积系统中泊松积分器修正哈密顿量的重要性，以洛特卡-沃尔泰拉动力学为例，展示5维不可积洛特卡-沃尔泰拉系统并通过数值研究其哈密顿泊松积分器的稳定性性质。

Deep Learning: Hydrodynamics, and Lie-Poisson Hamilton-Jacobi Theory

The interpretation of deep learning as a dynamical system has gained a considerable attention in recent years as it provides a promising framework. It allows for the use of existing ideas from established fields of mathematics for studying deep neural networks. In this article we present deep learning as an equivalent hydrodynamics formulation which in fact possess a Lie-Poisson structure and this further allows for using Poisson geometry to study deep learning. This is possible by considering the training of a deep neural network as a stochastic optimal control problem, which is solved using …

用能量方法研究Poisson平衡和Hamilton相对平衡的稳定性

Stability of Poisson Equilibria and Hamiltonian Relative Equilibria by Energy Methods

我们发展了Poisson动力系统平衡点和对称性哈密顿系统相对平衡点的一般稳定性理论，包括能量-Casimir和能量-动量方法的几个推广。利用拓扑推广的李雅普诺夫的结果，极值临界点的保守量是稳定的，我们表明，泊松平衡是稳定的，如果它是一个孤立的点在相交的水平集的保守函数的一个子集的相空间，这是有关的非Hausdorff性质的辛叶空间在这一点上。这一标准适用于推广的能量动量方法的哈密顿系统，这是不变的非紧对称群的共伴随轨道空间是不是豪斯多夫。我们还证明了一个$G$ -稳定的相对平衡满足$A$ -稳定的更强条件，其中$A$是$G$的一个特定的群理论定义的子集，它包含相对平衡的动量稳定子群.