动力学与控制论力学系统的自由度
胡海岩
摘要: 力学系统的自由度定义源自描述系统位形的独立坐标数。在分析力学发展过程中，人们通过对非完整约束的研究，将其拓展为独立的坐标变分数。本文指出，对于含非完整约束的力学系统，该定义存在不妥之处，给出的自由度会过度限制系统的力学行为。文中研究力学系统在状态空间中的可达流形，指出可达流形维数与描述系统动力学的一阶常微分方程组的最少未知函数个数一致，例如 Gibbs-Appell 方程与广义速度方程联立的未知函数个数，进而将可达流形维数的一半定义为系统自由度。通过含黏弹性支承的振动系统、在倾斜平面上运动的冰橇等案例，讨论了单个非完整约束导致的半自由度概念，指出其力学意义和与相邻整数自由度的关系。此外，文中还给出两个非完整约束导致系统减少一个自由度的案例，讨论了系统的切丛和余切丛维数。
On the degrees of freedom of a mechanical system
Haiyan Hu
Abstract：The definition of degrees of freedom of a mechanical system originated from the number of independent coordinates to describe the system configuration. The definition turned to be the number of independent variations of generalized coordinates after the studies on non-hololomic constraints in the development of analytic mechanics. The paper points out that the above definition of degrees of freedom has some flaws for the mechanical system with non-holonomic constraints and may impose excessive limits on the system dynamics. The paper, hence, studies the accessible state manifold of a mechanical system with non-holonomic constraints in the state space and shows that the dimensions of the accessible state manifold is equal to the number of minimal unknown variables to describe the system dynamics, governed by a set of ordinary differential equations of the first order, such as the Gibbs-Appell equations together with the relation of generalized velocities and psudo-velocities. Then, the paper defines the degrees of freedom of a mechanical system as a half of the dimensions of the accessible state manifold. Afterwards, the paper demonstrates how to understand the concept of a half degree of freedom of a mechanical system with a single non-holonomic constraint via two case studies, that is, the vibration system having a viscoelastic mounting and the sleigh system moving on an inclined plane, presenting the relation between a half degree of freedom and the two neighboring integer degrees of freedom. Furthermore, the paper gives two examples of mechanical systems, each of which has two non-holonomic constraints and results in the reduction of a single degree of freedom, and addresses the dimensions of tangent and cotangent bundles of the two systems.
Keyword：degrees of freedom, half degree of freedom, non-holonomic constraint, accessible state manifold, tangent bundle, Gibbs-Appell equation, viscoelasticity
