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## 物理代写|分析力学代写Analytical Mechanics代考|Applications of Lagrange’s Equations

In order to write down Lagrange’s equations associated with a given mechanical system, the procedure to be followed is rather simple. First, generalised coordinates $q_1, \ldots, q_n$ must be chosen. Next, the kinetic and potential energies relative to an inertial reference frame must be expressed exclusively in terms of the $q$ s and $\dot{q}$ s, so that the Lagrangian $L=T-V$ is also expressed only in terms of the generalised coordinates and velocities. Finally, all one has to do is compute the relevant partial derivatives of $L$, insert them into Eqs. (1.100) and the process of constructing the equations of motion for the system is finished. Let us see some examples of this procedure.

Example1.17 A bead slides on a smooth straight massless rod which rotates with constant angular velocity on a horizontal plane. Describe its motion by Lagrange’s formalism.
Solution
Let $x y$ be the horizontal plane containing the rod and let us use polar coordinates to locate the bead of mass $m$ (see Fig. 1.7). The two variables $r, \theta$ cannot be taken as generalised coordinates because $\theta$ is restricted to obey $\theta-\omega t=0$, which is a holonomic constraint ( $\omega$ is the rod’s constant angular velocity, supposed known). The system has only one degree of freedom associated with the radial motion and we can choose $q_1=r$ as generalised coordinate. According to Example 1.16, the kinetic energy can be put in the form
$$T=\frac{m}{2}\left(\dot{r}^2+r^2 \dot{\theta}^2\right)=\frac{m}{2}\left(\dot{r}^2+\omega^2 r^2\right),$$
where $\dot{\theta}=\omega$ has been used. Setting the plane of motion as the zero level of the gravitational potential energy, the Lagrangian for the system reduces to the kinetic energy:
$$L=T-V=\frac{m}{2}\left(\dot{r}^2+\omega^2 r^2\right) .$$
Now that we have the Lagrangian expressed only in terms of $r$ and $\dot{r}$, the equation of motion for the system follows at once:
$$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{r}}\right)-\frac{\partial I}{\partial r}=0 \Longrightarrow \frac{d}{d t}(m \dot{r})-m \omega^2 r=0 \Longrightarrow \ddot{r}=\omega^2 r .$$
One concludes that the bead tends to move away from the rotation axis due to a “centrifugal force”, which is the well-known result.

## 物理代写|分析力学代写Analytical Mechanics代考|Generalised Potentials

If the generalised forces can be derived from a function $U\left(q_1, \ldots, q_n, \dot{q}_1, \ldots, \dot{q}_n, t\right)$ by means of the equations
$$Q_k=-\frac{\partial U}{\partial q_k}+\frac{d}{d t}\left(\frac{\partial U}{\partial \dot{q}_k}\right),$$
then Eqs. (1.83) still imply Eqs. (1.100) with the Lagrangian defined by
$$L=T-U .$$
The function $U$ is called a generalised potential or velocity-dependent potential. The set of forces encompassed by Eq. (1.132) is larger than the set of conservative forces, the latter corresponding to the particular case in which $U$ depends neither on the generalised velocities nor on the time. The inclusion of forces derivable from a generalised potential is not fruit of a desire to achieve the greatest mathematical generality without physical consequences, for the electromagnetic force on a moving charge can only be derived from a generalised potential. In virtue of its great importance, this topic deserves a detailed discussion.

# 分析力学代考

## 物理代写|分析力学代写分析力学代考|拉格朗日方程的应用

$$T=\frac{m}{2}\left(\dot{r}^2+r^2 \dot{\theta}^2\right)=\frac{m}{2}\left(\dot{r}^2+\omega^2 r^2\right),$$
，其中$\dot{\theta}=\omega$已被使用。设运动平面为重力势能的零级，系统的拉格朗日量化为动能:
$$L=T-V=\frac{m}{2}\left(\dot{r}^2+\omega^2 r^2\right) .$$

$$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{r}}\right)-\frac{\partial I}{\partial r}=0 \Longrightarrow \frac{d}{d t}(m \dot{r})-m \omega^2 r=0 \Longrightarrow \ddot{r}=\omega^2 r .$$

## 物理代写|分析力学代写解析力学代考|广义势

$$Q_k=-\frac{\partial U}{\partial q_k}+\frac{d}{d t}\left(\frac{\partial U}{\partial \dot{q}_k}\right),$$

$$L=T-U .$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
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