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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|分析力学代写Analytical Mechanics代考|d’Alembert’s Principle

We are interested in dynamics, which can be formally reduced to statics by writing Newton’s second law in the form $\mathbf{F}_i-\dot{\mathbf{p}}_i=0$, with $\mathbf{p}_i=m_i \dot{\mathbf{r}}_i$. According to d’Alembert’s interpretation, each particle in the system is in “equilibrium” under a resultant force, which is the real force plus an “effective reversed force” equal to $-\dot{\mathrm{p}}_i$. This fictitious additional force is an inertial force existent in the non-inertial frame that moves along with the particle – that is, in which it remains at rest (Sommerfeld, 1952; Lanczos, 1970). Interpretations aside, the fact is that now, instead of (1.55), the equation
$$\sum_i\left(\dot{\mathbf{p}}_i-\mathbf{F}_i\right) \cdot \delta \mathbf{r}_i=0$$
is obviously true no matter what the virtual displacements $\delta \mathbf{r}_i$ are. Using again the decomposition (1.54) and assuming the virtual work of the constraint forces vanishes, we are led to d’Alembert’s principle:
$$\sum_i\left(\dot{\mathbf{p}}_i-\mathbf{F}_i^{(a)}\right) \cdot \delta \mathbf{r}_i=0 .$$
This principle is an extension of the principle of virtual work to mechanical systems in motion. For constrained systems, d’Alembert’s principle is a substantial leap forward with respect to the Newtonian approach hecause it excludes any reference to the constraint forces. In concrete applications, however, one must take into account that the virtual displacements $\delta \mathbf{r}_i$ are not independent because they have to be in harmony with the constraints.

Example1.12 Use d’Alembert’s principle to find the equations of motion for the mechanical system of Fig. 1.5, known as Atwood’s machine.

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

Provided the system is holonomic, it is possible to introduce a certain number $n$ of independent variables, generically denoted by $q_1, \ldots, q_n$ and called generalised coordinates, such that: (a) the position vector of each particle is unambiguously determined at any instant by the values of the $q \mathrm{~s} ;$ (b) the constraints, assumed of the form (1.41), are identically satisfied if expressed in terms of the $q \mathrm{~s}$. Let us see two illustrative cases.

Example1.13 In the case of the plane double pendulum, defined in Example 1.4, a possible choice of generalised coordinates is $q_1=\theta_1, q_2=\theta_2$ (see Fig. 1.1). Then,
\begin{aligned} &x_1=l_1 \sin \theta_1, \quad y_1=l_1 \cos \theta_1, \ &x_2=l_1 \sin \theta_1+l_2 \sin \theta_2, \quad y_2=l_1 \cos \theta_1+l_2 \cos \theta_2 . \end{aligned}
Note that the values of $\theta_1$ and $\theta_2$ completely specify the positions of the particles that is, the system’s configuration. In terms of $\theta_1$ and $\theta_2$, the constraint equations (1.40) reduce to the identities $l_1^2 \sin ^2 \theta_1+l_1^2 \cos ^2 \theta_1-l_1^2 \equiv 0$ and $l_2^2 \sin ^2 \theta_2+l_2^2 \cos ^2 \theta_2-l_2^2 \equiv 0$.
Example 1.14 A particle is restricted to the surface of a sphere in uniform motion. Let $\mathbf{u}=\left(u_x, u_y, u_z\right)$ be the sphere’s constant velocity relative to an inertial reference frame. At instant $t$ the centre of the sphere has coordinates $\left(u_x t, u_y t, u_z t\right)$ and the constraint equation takes the form
$$\left(x-u_x t\right)^2+\left(y-u_y t\right)^2+\left(z-u_z t\right)^2-R^2=0,$$
where $R$ is the radius of the sphere. Introducing the angles $\theta$ and $\phi$ by means of the equations
$$x=u_x t+R \sin \theta \cos \phi, y=u_y t+R \sin \theta \sin \phi, z=u_z t+R \cos \theta,$$ the constraint equation is now identically satisfied. Therefore, $q_1=\theta$ and $q_2=\phi$ is a possible choice of generalised coordinates.

# 分析力学代考

## 物理代写|分析力学代写分析力学代考|d ‘Alembert原理

$$\sum_i\left(\dot{\mathbf{p}}_i-\mathbf{F}_i\right) \cdot \delta \mathbf{r}_i=0$$

$$\sum_i\left(\dot{\mathbf{p}}_i-\mathbf{F}_i^{(a)}\right) \cdot \delta \mathbf{r}_i=0 .$$

## 物理代写|分析力学代写解析力学代考|广义坐标

\begin{aligned} &x_1=l_1 \sin \theta_1, \quad y_1=l_1 \cos \theta_1, \ &x_2=l_1 \sin \theta_1+l_2 \sin \theta_2, \quad y_2=l_1 \cos \theta_1+l_2 \cos \theta_2 . \end{aligned}

$$\left(x-u_x t\right)^2+\left(y-u_y t\right)^2+\left(z-u_z t\right)^2-R^2=0,$$
，其中$R$是球的半径。通过
$$x=u_x t+R \sin \theta \cos \phi, y=u_y t+R \sin \theta \sin \phi, z=u_z t+R \cos \theta,$$引入角$\theta$和$\phi$，约束方程现在同满足。因此，$q_1=\theta$和$q_2=\phi$可能是广义坐标的选择

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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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