标题： 哈密顿动力系统在理性力学中的几何场论 显示英文标题

标题： Geometrical Field Theory of Hamilton Dynamic System In Rational Mechanics

摘要： 当一组粒子在势场中运动时，有两个方面需要关注：1）粒子在空间域中的相对运动；2）粒子在时间域中的速度变化。 处理该系统的主要困难源于时间域中的运动和空间域中的运动完全耦合在一起的事实。 一般来说，对于由一组广义速度函数建立的哈密顿动力系统，已经建立了几种抽象理论，如李代数、辛流形、泊松括号等。 然而，从数学上讲，即使对于非常简单的问题，找出一个通用的哈密顿函数也是非常困难的。 受这些抽象数学研究的启发，通过变形的几何场理论来研究哈密顿动力系统。 首先，参考瞬时构型，在空间域中建立变形张量，在时间域中建立速度变换张量，针对由一组广义速度函数定义的动力系统。 其次，获得了速度空间域中的通用变形张量。 从连续介质力学的观点来看，通过引入空间的背景特征来定义应力张量。 然后，建立了通用的运动方程。 基于这些方程，简单的运动被分为两类：1）稳定运动；2）辐射运动。 它们都有量子解结构。 研究时空连续体的特征，以获得对基本物理事实的一些内在理解。 这项研究展示了一种处理哈密顿动力系统的工程方法。 显示英文摘要

摘要： When a set of particles are moving in a potential field, two aspects are concerned: 1) the relative motion of particle in spatial domain; 2) the particle velocity variations in time domain. The difficulty on treating the systems is originated from the fact that the motion in time domain and the motion in spatial domain are coupled together completely. Generally, for a Hamilton dynamic system established by a set of general velocity functions, several abstract theories have been well established, such as Lie algebra, Symplectic manifold, Poisson brackets, and others. However, mathematically, to find out a general Hamilton function is very difficult even for very simple problems. Inspired by these abstract mathematic researches, the Hamilton dynamic system is studied by geometrical field theory of deformation. Firstly, referring to the instant configuration, the deformation tensor in spatial domain and the velocity transformation tensor in time domain are established for a dynamic system defined by a set of general velocity functions. Secondly, the general deformation tensor in velocity space domain is obtained. From continuum mechanics point, the stress tensor is defined through introducing background feature of space. Then, the general motion equations are established. Based on them, the simple motions are divided into two classes: 1) stable motion; and 2) radiating motion. Both of them have quantum solution structures. The features of space-time continuum are studied to obtain some intrinsic understanding about basic physical facts. This research shows an engineering way to treat the Hamilton dynamic system.