从理论力学角度看翻转陀螺——翻转陀螺的部分定性结论

正2019年亚洲物理奥林匹克竞赛(APho)第3题考察了翻转陀螺。考生们需要利用刚体定点转动的知识对翻转陀螺进行建模,面临以下两个难点。首先,考生们没有系统的学过刚体定点转动的知识,对诸如转动惯量张量、欧拉角等基本概念不了解。其次,求解过程当中涉及地面参考系、质心参考系和转动参考系之间的变换。学生们需要区分同一个物理量,例如角速度,在不同参考系当中的表达形式以及它的运动方程。     

刚体一般运动的描述

刚体的一般运动是刚体运动学中最复杂的一类运动,其求解通常需要借助欧拉定理或沙勒定理.通过这两个定理,我们可以把刚体的一般运动分解成较简单的定轴转动和平动.本文主要应用代数理论中的正交矩阵描述刚体的运动,并用代数语言分析了定点转动的本征问题,证明了欧拉定理.随后,将刚体的定点转动进行分解,并给出了物理图像和推导结论,完成了对刚体复杂的一般运动的简单描述.     

刚体定点转动的欧拉定理

刚体定点转动是一个著名的力学问题．在Ｈ．Ｇｏｌｄｓｔｅｉｎ所著的《经典力学》一书中，有一段内容利用矩阵论和代数方程论论述了这个问题的欧拉定理［１］．本文在同一数学框架中讨论这个问题，而比文献［１］的论述显得简要；另外，还严格证明了描述刚体定点转动的转动（变换）矩阵的复特征值之幅角就是欧拉定理中的旋转角     

转动张量并矢式及与之相关的一些讨论

由转动张量对轴的不变性,应用转动张量的并矢式可推导出转动张量的几种具体形式,以及梳理刚体有限转动和无限小转动的数理逻辑关系,并由此给出了无限小转动对易性的解释。最后,对文献上一些关于刚体有限转动和无限小转动的讨论进行再讨论。     

刚体运动三维可视化模拟和复摆相关问题探讨

本文应用刚体定轴转动定律和角动量定理,结合VPython软件,首先模拟刚体的定轴转动,例如匀质圆柱体在不同力矩作用下的转动,直观地显示刚体运动轨迹以及角位移、角速度等物理量随时间的变化关系,以及重力矩作用下的刚体能量守恒等,使刚体运动过程实现可视化.有趣的是,本文还研究了刚体的平面平行运动(平动加转动)以及刚体的振动,模拟了大角度下刚体摆动时出现的混沌状态.可视化模拟深化了对课本知识的理解.     

刚体力学教学讨论两则

本文包括对刚体力学中两个教学问题的讨论：（１）关于有限转动不矢量的论证；指出论证中所用的有限转动的转轴不应是空间固定轴，而应是本体固定轴。（２）刚体定点转动的自转轴和转动瞬轴；分析了两者在物理意义上的不同。     

对《理论力学教程》中一个问题的再讨论

关于"一张错误的插图"问题,文献[1-3]对经典教材《理论力学教程》中描述刚体平面平行运动的公式(3.7.3)的理解有误.教材中公式(3.7.3)是刚体上任一点的速度在动坐标系中的分量形式.由于对坐标系与参考系概念的区分不清楚,参考文献[1-3]的作者以及部分学生和教师认为动坐标系是在转动参考系中建立的.本文作者认为研究刚体平面平行运动的3.7节没有选择转动参考系;为了研究的方便,以地面为参考系中建立了两个坐标系来描述刚体的平面平行运动.     

非完整映射理论与刚体定点转动的几何描述

利用非完整映射方法,从一个已知Riemann空间构造一个嵌入其中的Riemann-Cartan空间.作为特例,研究从Euclidean空间构造Weitzenbock空间的方法.基于dAlembert-Lagrange原理和非完整映射,将一个Riemann空间的测地线对应于另一个Riemann-Cartan空间的自平行线.把这种非完整映射理论应用到刚体定点转动问题上,得到了刚体运动的欧拉方程是欧拉角描述的Riemann位形空间的测地线方程,而在刚体角速度对应的准坐标空间上是常挠率Riemann-Cartan空间的自平行线方程的结论. 关键词： 欧拉角 非完整映射 Riemann-Cartan空间 自平行线     

一般运动刚体的瞬时螺旋轴

本文导出了一般运动刚体的速度最小值、速度量小值点的位矢、速度最小值点加速度的数学表达式．通过这些公式的讨论验证了一般运动刚体瞬时螺旋轴的存在，给出了清晰的物理图象和几何图象；把潘索定理推广到刚体的一般运动情形，给出了一般运动刚体上任一点速度的又一求法．作为其特例讨论了刚体的定点转动和平面平行运动．     

刚体转动方程的矢量式

就刚体动力学中角动量公式和转动方程进行了分析, 指出转动惯量张量不是常量, 得到了相关的矢量 式, 并分析了一些对公式L=J ω 和M =J β的常见误解     

连续变形体定点转动实例的计算机解

连续变形体定点转动问题广泛地用于体育力学及空间飞行器力学中,它是刚体定点转动问題的延拓,本文从角动量定理的一般形式导出连续变形体定点转动的一般解析表达式。这一表达式未曾见于一般力学文献中。可以看出,这是一组变系数的非线性方程组。应用电子计算机可以方便地求得给定条件下这组方程的数值解和运动的图形表示。文中通过两个实例说明这组方程的某些应用。     

Integrable cases of a rigid body dynamics and integrable systems on the ellipsoids

Infinite-dimensional sets of integrable cases are found for the equations of a rigid body rotation around a fixed point in an axially symmetric potential field and also in more complicated fields in the presence of some symmetry of the rigid body inertia tensor.     

“自然坐标系”下的角动量定理及其应用

阐述在刚体的定点转动中,引入＂自然坐标系＂来分析,往往可以使问题变得更加简单,物理意义变得更加清晰,并且通过具体实例加以验证,最后通过对刚体定轴转动的分析,把刚体定轴转动的角动量定理与定点的角动量定理统一起来.     

Complete list of first integrals of dynamic equations of motion of a 4D rigid body in a nonconservative field under the assumption of linear damping

The dynamic part of equations of motion of a dynamically symmetric 4D rigid body, where the force field is concentrated on that part of the body that has the form of a two-dimensional disc, is investigated. In this case, the tensor of the angular velocity of such a body is six-dimensional, while the velocity of the center of mass is four-dimensional. Under certain conditions, a complete list of first integrals, which are expressed through elementary functions, is obtained.     

On the geometry of motions in one integrable problem of the rigid body dynamics

Due to Poinsot’s theorem, the motion of a rigid body about a fixed point is represented as rolling without slipping of the moving hodograph of the angular velocity over the fixed one. If the moving hodograph is a closed curve, visualization of motion is obtained by the method of P.V. Kharlamov. For an arbitrary motion in an integrable problem with an axially symmetric force field the moving hodograph densely fills some two-dimensional surface and the fixed one fills a three-dimensional surface. In this paper, we consider the irreducible integrable case in which both hodographs are two-frequency curves. We obtain the equations of bearing surfaces, illustrate the main types of these surfaces. We propose a method of the so-called non-straight geometric interpretation representing the motion of a body as a superposition of two periodic motions.     

力矩突变时角动量变化规律的实验演示与分析

给出一种演示刚体在定点运动时角动量与角速度方向的关系,及力矩突变为零时角动量变化规律的实验演示与分析.     

Application of stereo vision to the study of mixed-mode crack-tip deformations

A stereo vision is applied to evaluate crack-tip parameters for fracture specimens subjected to a mixed-mode loading (tension and shear). By using a special loading device, the applied remote loading is oriented at an angle with respect to the axis of the crack. At each loading angle, the calibrated vision system was rotated so that the axis of the crack is parallel to the horizontal or vertical axis of the image frame. At a load close to the crack initiation, the displacement field around the crack-tip region of the fracture specimen was measured relative to a specimen coordinate system located at the crack tip of the fracture specimen. During the experiment, the fracture specimen was subjected to rigid body translation and rotation. Hence, the displacement fields are affected by the rigid body translation and rotation. Using the experimentally determined displacements and the analytically determined displacements with several higher order terms being included, the stress intensity factors and the amount of rigid body translation and rotation were calculated through a least-squares fit. The effect of the rigid body motion on the measured displacements was then eliminated using the computed rigid body translation and rotation. Experimental results indicate that a K_I and K_II dominant region is observed in the corrected displacement fields.     

Exact Solution to the Schrödinger Equation for the Quantum Rigid Body

The exact solution to the Schrödinger equation for the rigid body with the given angular momentum and parity is obtained. Since the quantum rigid body can be thought of as the simplest quantum three-body problem where the internal motion is frozen, this calculation method is a good starting point for solving the quantum three-body problems.