2022年周衍柏《理论力学》第五章教案-分析力学 .pdf

第五章分析力学本章要求（1）掌握分析力学中的一些基本概念；（2）掌握虚功原理；（3）掌握拉格朗日方程；（4）掌握哈密顿正则方程。第一节约束和广义坐标一、约束的概念和分类 加于力学体系的限制条件叫约束。按不同的标准有不同的分类：按约束是否与时间有关分类：稳定约束、不稳定约束；按质点能否脱离约束分类：可解约束、不可解约束；按约束限制范围分类：几何约束（完整约束）、运动约束（不完整约束）。本章只讨论几何约束（完整约束），这种约束下的体系叫完整体系。二、广义坐标 1、自由度描述一个力学体系所需要的独立坐标的个数叫体系的自由度。设体系有n 个粒子，一个粒子需要3 个坐标（如 x、y、z）描述，而体系受有K个约束条件，则体系的自由度为（3n-K）2、广义坐标描述力学体系的独立坐标叫广义坐标。例如：作圆周运动的质点只须角度用描述，广义坐标为，自由度为1，球面上运动的质点，由极角和描述，自由度为2。第二节虚功原理本节重点要求：掌握虚位移、虚功、理想约束等概念；掌握虚功原理。一、实位移与虚位移质点由于运动实际上所发生的位移叫实位移；在某一时刻，在约束允许的情况下，质点可能发生的位移叫虚位移。如果约束为固定约束，则实位移是虚位移中一的个；若约束不固定，实位移与虚位移无共同之处。例如图5.2.1 中的质点在曲面上运动，而曲面也在移动，显然实位移与虚位移不一致。二、理想约束设质点系受主动力和约束力的作用，它们在任意虚位移中作的功叫虚功。若约束反力在任意虚位移中对质点系所作虚功之和为零，则这种约束叫理想约束。光滑面、光滑线、刚性杆、不可伸长的绳等都是理想约束。三、虚功原理 1、文字叙述和数学表示：受理想约束的力学体系，平衡的充要条件是：作用于力学体系的诸主动力在任意虚位移中作的元功之和为零。即（1）适用条件：惯性系、理想不可解约束。2、推论设系统的广义坐标为q1，qa，qS，虚位移可写为用广义坐标变分表示的形式：定义：称为相应于广义坐标qa的广义力，则虚功原理表述为：理想约束的力学体系平衡的充要条件为质点系受的广义力为零，即：文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3（2）3、用虚功原理求解平衡问题的方法步骤一般步骤为：（1）确定自由度，选取坐标系，分析力（包括主动力、约束力）；（2）选取广义坐标并将各质点坐标表示成广义坐标qa的函数：；（3）求主动力的虚功并令其为零：，由此求出平衡条件。例 见书 P276 例 1第三节拉格朗日方程本节重点要求：（1）掌握拉格朗日方程的两种形式，方程的特点和适用条件等；（2）掌握用拉格朗日方程求解具体问题的步骤；（3）了解循环积分等概念。一、基本形式的拉格朗日方程 1、方程的推导由牛顿第二定律并应用理想约束的条件，可以得到达朗伯拉格朗日方程：(1)将坐标的变分改成用广义坐标q1，qS的变分表示，即：经数学运算，令（称为体系的动能），（称为相应于 qa的广义力），则（1）式变文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3为：（2）这就是基本形式的拉格朗日方程，应注意：（2）实际是一组方程。2、方程的适用条件：理想约束。二、保守系的拉格朗日方程设作用于体系的力全为保守力，则广义力可由（V为势能）求得：在普遍形式的拉氏方程（2）中，由于 V不包含广义速度，可令：（动能与势能的差）为拉格朗日函数，则（2）式变为：（3）应指出（3）的适用条件为保守系，理想约束，且（3）应用很普遍。三、应用拉格朗日方程求解问题的步骤，例一般步骤：画草图，确定自由度 s 和广义坐标 qa；分析主动力，若为保守系，则求出势能 V；若为非保守力，则计算广义力 Qa；求动能 T=T（）；对保守系，求出 L=T-V，进而代入方程（3），写出运动方程；对非保守系，将 T和广义力 Q 代入方程（2），写出运动方程。解方程，求出q（t）。例 1 P265 4.10题圆环在光滑圆圈上运动，而圆圈绕垂直圆面的轴作匀角速运动，求圆环运动规律。文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3解：方法一：牛顿力学方法（已在第四章第三节作为举例计算）方法二：用拉格朗日方程求解。这是光滑圆圈且受的力只有重力和约束力，属于保守体系，可采用保守系的拉氏方程求解。质点自由度为1，转角为广义坐标，广义速度为。任一角度时圆环（视为质点）的动能，其中绝对速度 v 可由速度合成公式求出：这里（方向沿切线方向），牵连速度，大小为，方向垂直于 op。由速度合成公式得到：动能：取圆平面为零势能位置，则V=0，从而 L=T-V=T-0=T 代入拉氏方程（2）中：，得到四、循环积分。若拉氏函数L 中某一坐标 qi不出现，则该坐标qi叫循环坐标，则（常数），叫循环积分。第五节 哈密顿正则方程本节不作重点要求。基本要求是：了解正则坐标、正则动量的概念和正则方程及其应用。一、哈密顿函数 设力学体系的广义坐标为，广文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3义速度为，则拉格朗日函数,定义广义动量，则函数叫哈密顿函数。它是广义坐标、广义动量的函数，而广义坐标、广义动量称为正则变量。特例：对保守体系，H=T+V （动能与势能之和）二、哈密顿正则方程 哈密顿函数满足的方程为：由该方程组也可探讨运动规律。方程组（1）叫哈密顿正则方程。三、用哈密顿正则方程求解问题的步骤一般步骤为：确定自由度r 和广义坐标求动能 T和势能 V，写出拉格朗日函数。求广义动量，将 T和 V中的换为，写出 H=T+V=H（，）、写出正则方程，进而解方程。例电子的运动（见书P314-316）最后指出：拉格朗日方程和哈密顿正则方程都是分析力学中的基本方程，其作用与牛顿第二定律一样，其中拉氏方程为二阶微分方程，哈密顿正则方程为一阶微分方程，但个数比前者多一倍。文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3文档编码:CU1K2K7H4D10 HS10A1F6B3K8 ZY5S9D6X2H3