Geometrical Field Theory of Hamilton Dynamic System In Rational Mechanics

TL;DR

This paper introduces a geometrical field theory approach to Hamilton dynamic systems in rational mechanics, addressing coupled spatial and temporal motions through deformation tensors and defining motion classes with quantum structures.

Contribution

It develops a geometrical deformation-based framework for Hamilton systems, providing new equations and classifications of motions with quantum features.

Findings

Deformation tensors in spatial and velocity spaces are established.

Motion equations are derived from a continuum mechanics perspective.

Stable and radiating motions are classified with quantum solution structures.

Abstract

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.