  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.
