## 物理代写|分析力学代写Analytical Mechanics代考|Infinitesimal Translations and Rotations

statistics-lab™ 为您的留学生涯保驾护航 在代写分析力学Analytical Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写分析力学Analytical Mechanics代写方面经验极为丰富，各种代写分析力学Analytical Mechanics相关的作业也就用不着说。

• 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|Infinitesimal Translations and Rotations

Consider the infinitesimal transformation
$$\mathbf{r}i \rightarrow \mathbf{r}_i^{\prime}=\mathbf{r}_i+\delta \mathbf{r}_i, \quad \mathbf{v}_i \rightarrow \mathbf{v}_i^{\prime}=\mathbf{v}_i+\delta \mathbf{v}_i$$ where $\delta \mathbf{r}_i$ and $\delta \mathbf{v}_i$ are infinitesimal displacements of the positions and velocities of a system of $N$ particles. Let $$L\left(\mathbf{r}_1, \ldots, \mathbf{r}_N, \dot{\mathbf{r}}_1, \ldots, \dot{\mathbf{r}}_N, t\right) \equiv L(\mathbf{r}, \mathbf{v}, t)$$ be the Lagrangian for the system. The variation of $L$ under transformation (2.131) is defined by $$\delta L=L\left(\mathbf{r}^{\prime}, \mathbf{v}^{\prime}, t\right)-L(\mathbf{r}, \mathbf{v}, t)=\sum{i=1}^N\left[\frac{\partial L}{\partial \mathbf{r}_i} \cdot \delta \mathbf{r}_i+\frac{\partial L}{\partial \mathbf{v}_i} \cdot \delta \mathbf{v}_i\right]$$

where we have used the notation
$$\frac{\partial L}{\partial \mathbf{r}_i} \equiv \frac{\partial L}{\partial x_i} \hat{\mathbf{x}}+\frac{\partial L}{\partial y_i} \hat{\mathbf{y}}+\frac{\partial L}{\partial z_i} \hat{\mathbf{z}},$$
with a similar definition for $\partial L / \partial \mathbf{v}_i$.
Example 2.15 A rigid translation of the system of particles consists in the same displacement $\boldsymbol{\epsilon}=\epsilon \hat{\mathbf{n}}$ of all particles of the system, so that the velocities, as well as the relative positions of the particles, remain unchanged. In this case, equations (2.131) are valid with
$$\delta \mathbf{r}_i=\epsilon \hat{\mathbf{n}}, \quad \delta \mathbf{v}_i=0, \quad \text { (translation) }$$
where $\epsilon$ is an infinitesimal parameter with dimension of length.
Example 2.16 A rigid rotation of the system of particles consists in a rotation of all vectors of the system through the same angle $\delta \theta$ about an axis defined by the unit vector $\hat{\mathbf{n}}$. Inspecting Fig. 2.7 one infers that $\left|\delta \mathbf{r}_i\right|=r_i \sin \alpha \delta \theta$, which has the appearence of magnitude of a vector product. In fact, defining the vector $\delta \theta=\delta \theta \hat{\mathbf{n}}$, it follows at once that the correct expression for the vector $\delta \mathbf{r}_i$ is $\delta \mathbf{r}_i=\delta \theta \times \mathbf{r}_i$. Since each velocity $\delta \mathbf{v}_i$ undergoes the same rotation, equations (2.131) are valid with
$$\delta \mathbf{r}_i=\delta \theta \times \mathbf{r}_i, \quad \delta \mathbf{v}_i=\delta \theta \times \mathbf{v}_i, \quad \text { (rotation) }$$
where $\delta \theta$ is the infinitesimal parameter associated with the transformation.

## 物理代写|分析力学代写Analytical Mechanics代考|Conservation Theorems

In order to state the results in a sufficiently general form, let us assume that the system described by the Lagrangian (2.132) is subject to the holonomic constraints
$$f_s\left(\mathbf{r}1, \ldots, \mathbf{r}_N, t\right)=0, \quad s=1, \ldots, p .$$ Theorem 2.5.2 Let a mechanical system be described by the Lagrangian $L=T-V$, where $V$ is a velocity-independent potential. If the Lagrangian and the constraints (2.137) are invariant under an arbitrary rigid translation, then the total linear momentum of the system is conserved. Proof According to the discussion at the end of Section 2.4, the Lagrangian $$\mathcal{L}=L(\mathbf{r}, \mathbf{v}, t)+\sum{s=1}^p \lambda_s f_s(\mathbf{r}, t)$$
yields, in one fell swoop, the equations of motion and the constraint equations. Since $L$ and the constraint equations are invariant under arbitrary translations, so is $\mathcal{L}$. In particular, $\mathcal{L}$ is invariant under any infinitesimal translation and we have
$$0=\delta \mathcal{L}=\sum_{i=1}^N \frac{\partial \mathcal{L}}{\partial \mathbf{r}i} \cdot(\epsilon \hat{\mathbf{n}})=\epsilon \hat{\mathbf{n}} \cdot \sum{i=1}^N \frac{\partial \mathcal{L}}{\partial \mathbf{r}i},$$ where we have used (2.133) and (2.135). Since $\epsilon$ is arbitrary, we obtain $$\hat{\mathbf{n}} \cdot \sum{i=1}^N \frac{\partial \mathcal{L}}{\partial \mathbf{r}_i}=0$$
Noting that
$$\frac{\partial \mathcal{L}}{\partial \mathbf{v}_i} \equiv \frac{\partial \mathcal{L}}{\partial \dot{\mathbf{r}}_i}=\frac{\partial L}{\partial \dot{\mathbf{r}}_i}=m_i \dot{\mathbf{r}}_i=\mathbf{p}_i,$$
where $\mathbf{p}_i$ is the linear momentum of the $i$ th particle, Lagrange’s equations can be written in the form
$$\frac{\partial \mathcal{L}}{\partial \mathbf{r}_i}=\frac{d}{d t}\left(\frac{\partial \mathcal{L}}{\partial \dot{\mathbf{r}}_i}\right)=\frac{d \mathbf{p}_i}{d t}$$

# 分析力学代考

## 物理代写|分析力学代写Analytical Mechanics代考|Infinitesimal Translations and Rotations

$$\mathbf{r} i \rightarrow \mathbf{r}_i^{\prime}=\mathbf{r}_i+\delta \mathbf{r}_i, \quad \mathbf{v}_i \rightarrow \mathbf{v}_i^{\prime}=\mathbf{v}_i+\delta \mathbf{v}_i$$

$$L\left(\mathbf{r}_1, \ldots, \mathbf{r}_N, \dot{\mathbf{r}}_1, \ldots, \dot{\mathbf{r}}_N, t\right) \equiv L(\mathbf{r}, \mathbf{v}, t)$$

$$\delta L=L\left(\mathbf{r}^{\prime}, \mathbf{v}^{\prime}, t\right)-L(\mathbf{r}, \mathbf{v}, t)=\sum i=1^N\left[\frac{\partial L}{\partial \mathbf{r}_i} \cdot \delta \mathbf{r}_i+\frac{\partial L}{\partial \mathbf{v}_i} \cdot \delta \mathbf{v}_i\right]$$

$$\frac{\partial L}{\partial \mathbf{r}_i} \equiv \frac{\partial L}{\partial x_i} \hat{\mathbf{x}}+\frac{\partial L}{\partial y_i} \hat{\mathbf{y}}+\frac{\partial L}{\partial z_i} \hat{\mathbf{z}}$$

$$\delta \mathbf{r}_i=\epsilon \hat{\mathbf{n}}, \quad \delta \mathbf{v}_i=0, \quad(\text { translation })$$

$$\delta \mathbf{r}_i=\delta \theta \times \mathbf{r}_i, \quad \delta \mathbf{v}_i=\delta \theta \times \mathbf{v}_i, \quad \text { (rotation) }$$

## 物理代写|分析力学代写Analytical Mechanics代考|Conservation Theorems

$$f_s\left(\mathbf{r} 1, \ldots, \mathbf{r}N, t\right)=0, \quad s=1, \ldots, p .$$ 定理 2.5.2 让机械系统用拉格朗日量描述 $L=T-V$ ，在哪里 $V$ 是与速度无关的势能。如果拉格朗日量 和约束 (2.137) 在任意刚性平移下不变，则系统的总线性动量守恒。证明 根据 2.4 节末尾的讨论，拉格朗 日量 $$\mathcal{L}=L(\mathbf{r}, \mathbf{v}, t)+\sum s=1^p \lambda_s f_s(\mathbf{r}, t)$$ 一举得出运动方程和约束方程。自从 $L$ 并且约束方程在任意平移下是不变的，因此是 $\mathcal{L}$. 尤其， $\mathcal{L}$ 在任何 无穷小平移下都是不变的，我们有 $$0=\delta \mathcal{L}=\sum{i=1}^N \frac{\partial \mathcal{L}}{\partial \mathbf{r} i} \cdot(\epsilon \hat{\mathbf{n}})=\epsilon \hat{\mathbf{n}} \cdot \sum i=1^N \frac{\partial \mathcal{L}}{\partial \mathbf{r} i},$$

$$\hat{\mathbf{n}} \cdot \sum i=1^N \frac{\partial \mathcal{L}}{\partial \mathbf{r}_i}=0$$

$$\frac{\partial \mathcal{L}}{\partial \mathbf{v}_i} \equiv \frac{\partial \mathcal{L}}{\partial \dot{\mathbf{r}}_i}=\frac{\partial L}{\partial \dot{\mathbf{r}}_i}=m_i \dot{\mathbf{r}}_i=\mathbf{p}_i$$

$$\frac{\partial \mathcal{L}}{\partial \mathbf{r}_i}=\frac{d}{d t}\left(\frac{\partial \mathcal{L}}{\partial \dot{\mathbf{r}}_i}\right)=\frac{d \mathbf{p}_i}{d t}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|分析力学代写Analytical Mechanics代考|Holonomic Constraints and Lagrange Multipliers

statistics-lab™ 为您的留学生涯保驾护航 在代写分析力学Analytical Mechanics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写分析力学Analytical Mechanics代写方面经验极为丰富，各种代写分析力学Analytical Mechanics相关的作业也就用不着说。

• 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|Holonomic Constraints and Lagrange Multipliers

Although equations of the form (2.75) do not include the most general non-holonomic constraints, they do include the holonomic ones. Indeed, holonomic constraints take the form
$$f_i(q, t)=0, \quad l=1, \ldots, p$$
hence, by taking the total time derivative,
$$\sum_{k=1}^n \frac{\partial f_i}{\partial q_k} \dot{q}_k+\frac{\partial f_l}{\partial t}=0, \quad l=1, \ldots, p$$

These equations are of the form (2.75) with
$$a_{l k}=\frac{\partial f_l}{\partial q_k}, \quad a_{l t}=\frac{\partial f_l}{\partial t} .$$
Thus, the Lagrange multiplier method can be applied to holonomic systems when it is inconvenient to replace the $q$ s by a smaller set of independent variables or when one wishes to determine the constraint forces. By way of illustration, let us apply the Lagrange multiplier formalism to a holonomic case.

In the holonomic case it is possible to choose generalised coordinates in terms of which the constraint equations are identically satisfied. Nevertheless, in the discussion of certain general theoretical questions it is sometimes advantageous to use an excessive number of coordinates and treat the holonomic constraints by the Lagrange multiplier technique. Given the holonomic constraints (2.96), consider the new Lagrangian $\mathcal{L}$ defined by
$$\mathcal{L}=L(q, \dot{q}, t)+\sum_{l=1}^p \lambda_l f_l(q, t) .$$
Introducing the $n+p$ variables $\xi_1, \ldots, \xi_{n+p}$ defined by
$$\xi_1=q_1, \ldots, \xi_n=q_n, \xi_{n+1}=\lambda_1, \ldots, \xi_{n+p}=\lambda_p$$
and treating them as $n+p$ independent generalised coordinates, the equations of motion and the constraint equations can be written in one fell swoop as
$$\frac{d}{d t}\left(\frac{\partial \mathcal{L}}{\partial \dot{\xi}_k}\right)-\frac{\partial \mathcal{L}}{\partial \xi_k}=0, \quad k=1, \ldots, n+p$$

## 物理代写|分析力学代写Analytical Mechanics代考|Constants of the Motion

A constant of the motion is a conserved physical quantity – that is, a quantity associated with a mechanical system whose value does not change during the dynamical evolution of the system. Mathematically, a constant of the motion, an invariant, a first integral or simply an integral of a mechanichal system is a function $f$ of the coordinates, velocities and, possibly, time that remains constant throughout the motion of the system:
$$\frac{d f}{d t}=0 \quad \text { or } \quad f\left(q_1(t), \ldots, q_n(t), \dot{q}_1(t), \ldots, \dot{q}_n(t), t\right)=\text { constant }$$
where the functions $q_k(t)$ satisfy the equations of motion for the system.
Example 2.10 In the case of a harmonic oscillator with angular frequency $\omega$,
$$f(x, \dot{x}, t)=\arctan \left(\frac{\omega x}{\dot{x}}\right)-\omega t$$
is a constant of the motion. In order to check this claim we have to compute the total time derivative of $f$, which is given by
$$\frac{d f}{d t}=\frac{\partial f}{\partial x} \frac{d x}{d t}+\frac{\partial f}{\partial \dot{x}} \frac{d \dot{x}}{d t}+\frac{\partial f}{\partial t}=\frac{1}{1+\omega^2 x^2 / \dot{x}^2} \frac{\omega}{\dot{x}} \dot{x}+\frac{1}{1+\omega^2 x^2 / \dot{x}^2}\left[-\frac{\omega x}{\dot{x}^2}\right] \ddot{x}-\omega .$$
Using the oscillator equation of motion, $\ddot{x}=-\omega^2 x$, Eq. (2.122) reduces to
$$\frac{d f}{d t}=\omega \frac{1+\omega^2 x^2 / \dot{x}^2}{1+\omega^2 x^2 / \dot{x}^2}-\omega=\omega-\omega=0$$
and $f$ is indeed a constant of the motion.
Constants of the motion supply first-order differential equations that give important information about the time evolution of the system. Even though the exact solution to the equations of motion may be unknown, the knowledge of certain first integrals often reveals physically relevant facts regarding the nature of the motion. Sometimes it is possible to obtain exact answers to certain questions thanks to the constants of the motion, without having to completely solve the dynamical problem. Therefore, investigating general conditions that ensure the existence of constants of the motion is highly relevant. The first and simplest result of such an investigation will be stated after two useful definitions.

# 分析力学代考

## 物理代写|分析力学代写Analytical Mechanics代考|Holonomic Constraints and Lagrange Multipliers

$$f_i(q, t)=0, \quad l=1, \ldots, p$$

$$\sum_{k=1}^n \frac{\partial f_i}{\partial q_k} \dot{q}k+\frac{\partial f_l}{\partial t}=0, \quad l=1, \ldots, p$$ 这些方程的形式为 (2.75)，其中 $$a{l k}=\frac{\partial f_l}{\partial q_k}, \quad a_{l t}=\frac{\partial f_l}{\partial t}$$

$$\mathcal{L}=L(q, \dot{q}, t)+\sum_{l=1}^p \lambda_l f_l(q, t)$$

$$\xi_1=q_1, \ldots, \xi_n=q_n, \xi_{n+1}=\lambda_1, \ldots, \xi_{n+p}=\lambda_p$$

$$\frac{d}{d t}\left(\frac{\partial \mathcal{L}}{\partial \dot{\xi}_k}\right)-\frac{\partial \mathcal{L}}{\partial \xi_k}=0, \quad k=1, \ldots, n+p$$

## 物理代写|分析力学代写Analytical Mechanics代考|Constants of the Motion

$$\frac{d f}{d t}=0 \quad \text { or } \quad f\left(q_1(t), \ldots, q_n(t), \dot{q}_1(t), \ldots, \dot{q}_n(t), t\right)=\text { constant }$$

$$f(x, \dot{x}, t)=\arctan \left(\frac{\omega x}{\dot{x}}\right)-\omega t$$

$$\frac{d f}{d t}=\frac{\partial f}{\partial x} \frac{d x}{d t}+\frac{\partial f}{\partial \dot{x}} \frac{d \dot{x}}{d t}+\frac{\partial f}{\partial t}=\frac{1}{1+\omega^2 x^2 / \dot{x}^2} \frac{\omega}{\dot{x}} \dot{x}+\frac{1}{1+\omega^2 x^2 / \dot{x}^2}\left[-\frac{\omega x}{\dot{x}^2}\right] \ddot{x}-\omega$$

$$\frac{d f}{d t}=\omega \frac{1+\omega^2 x^2 / \dot{x}^2}{1+\omega^2 x^2 / \dot{x}^2}-\omega=\omega-\omega=0$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## PHYS3318 Analytical Mechanics课程简介

Forbidden Overlap: Students may not receive credit for both PHYS 3314 and PHYS 3318.
Prerequisite: strong performance in PHYS 2218 or permission of instructor; AEP 4210 or appropriate course(s) in mathematics. Intended for students with exceptional background in physics/math. PHYS 3314 covers similar material, while assuming less prior knowledge. Makes use of Fourier analysis, linear differential equations, linear algebra, and vector analysis.

## PREREQUISITES

Covers Newtonian mechanics of particles and systems of particles, including rigid bodies, oscillating systems, gravitation and planetary motion, moving coordinate systems, Euler’s equations, Lagrange and Hamilton formulations, normal modes and small vibrations, and perturbation theory. At the level of Classical Mechanics by Goldstein, Mechanics by Landau and Lifshitz, and Analytical Mechanics by Hand and Finch.

## PHYS3318 Analytical Mechanics HELP（EXAM HELP， ONLINE TUTOR）

Theorem 2.3.1 Equivalent Lagrangians give rise to the same equations of motion.

Proof The action $\bar{S}$ associated with $\bar{L}$ is
$$\bar{S}=\int_{t_1}^{t_2} \bar{L}(q, \dot{q}, t) d t=\int_{t_1}^{t_2} L(q, \dot{q}, t) d t+\int_{t_1}^{t_2} \frac{d f}{d t} d t=S+f\left(q\left(t_2\right), t_2\right)-f\left(q\left(t_1\right), t_1\right)$$
Since the variation of the action leaves the endpoints $q\left(t_1\right)$ and $q\left(t_2\right)$ fixed, $\delta \bar{S}=\delta S$. Therefore, the conditions $\delta \bar{S}=0$ and $\delta S=0$ are identical, showing that $\bar{L}$ and $L$ engender exactly the same equations of motion.

Theorem 2.3.2 If $V: \mathbb{R} \rightarrow \mathbb{R}$ is a twice continuously differentiable function such that $V^{\prime \prime} \neq 0$ and the time interval $\left[t_1, t_2\right]$ is sufficiently short, the action
$$S[x]=\int_{t_1}^{t_2}\left[\frac{m}{2} \dot{x}(t)^2-V(x(t))\right] d t$$
is a local minimum for the physical path as compared to all other sufficiently close neighbouring paths with fixed endpoints $x\left(t_1\right)$ and $x\left(t_2\right)$.

Proof We follow Gallavotti $(1983)$. Let $\left[t_1, t_2\right]$ be a time interval so short that a single physical path goes through the fixed endpoints $x\left(t_1\right)$ and $x\left(t_2\right)$. Let $\eta:\left[t_1, t_2\right] \rightarrow \mathbb{R}$ be an infinitely differentiable function such that $\eta\left(t_1\right)=\eta\left(t_2\right)=0$. The physical path shall be denoted simply by $x(t)$. The action for the varied path $\bar{x}=x+\eta$ is
$$S[\bar{x}]=\int_{t_1}^{t_2}\left[\frac{m}{2}(\dot{x}+\dot{\eta})^2-V(x+\eta)\right] d t$$
By Taylor’s theorem with Lagrange’s form of the remainder (Spivak, 1994) we can write
$$V(x+\eta)=V(x)+V^{\prime}(x) \eta+\frac{1}{2} V^{\prime \prime}(x+\xi) \eta^2, \quad|\xi| \leq|\eta| .$$
Since we are only interested in establishing that the action is a local minimum for the physical path, let us choose $\eta$ such that $|\eta(t)| \leq 1$ for all $t \in\left[t_1, t_2\right]$, which implies $|\xi| \leq 1$. Substituting (2.57) into $(2.56)$, we are led to
\begin{aligned} S[\bar{x}] & =\int_{t_1}^{t_2}\left[\frac{m}{2} \dot{x}^2+m \dot{x} \dot{\eta}+\frac{m}{2} \dot{\eta}^2-V(x)-V^{\prime}(x) \eta-\frac{1}{2} V^{\prime \prime}(x+\xi) \eta^2\right] d t \ & =S[x]+\int_{t_1}^{t_2}\left[m \dot{x} \dot{\eta}-V^{\prime}(x) \eta\right] d t+\int_{t_1}^{t_2}\left[\frac{m}{2} \dot{\eta}^2-\frac{1}{2} V^{\prime \prime}(x+\xi) \eta^2\right] d t . \end{aligned}
Integrating by parts the term containing $\dot{\eta}$ and taking into account that the physical path obeys the equation of motion $m \ddot{x}+V^{\prime}(x)=0$, we obtain
$$S[\bar{x}]-S[x]=\int_{t_1}^{t_2}\left[\frac{m}{2} \dot{\eta}^2-\frac{1}{2} V^{\prime \prime}(x+\xi) \eta^2\right] d t$$

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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## 物理代写|分析力学代写Analytical Mechanics代考|PHY225

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## 物理代写|分析力学代写Analytical Mechanics代考|Virtual Work

The importance of introducing the notion of virtual displacement stems from the following observation: if the surface to which the particle is confined is ideally smooth, the contact force of the surface on the particle, which is the constraint force, has no tangential component and therefore is normal to the surface. Thus, the work done by the constraint force as the particle undergoes a virtual displacement is zero even if the surface is in mótion, differently from thé work donne during a reeal displaccement, which does nôt necessarily vanish. In most physically interesting cases, the total virtual work of the constraint forces is zero, as the next examples attest.

Example 1.10 Two particles joined by a rigid rod move in space. Let $\mathbf{f}_1$ and $\mathbf{f}_2$ be the constraint forces on the particles. By Newton’s third law $\mathbf{f}_1=-\mathbf{f}_2$ with both $\mathbf{f}_1$ and $\mathbf{f}_2$ parallel to the line connecting the particles. The virtual work done by the constraint forces is
$$\delta W_v=\mathbf{f}_1 \cdot \delta \mathbf{r}_1+\mathbf{f}_2 \cdot \delta \mathbf{r}_2=\mathbf{f}_2 \cdot\left(\delta \mathbf{r}_2-\delta \mathbf{r}_1\right) .$$
Setting $\mathbf{r}=\mathbf{r}_2-\mathbf{r}_1$, the constraint equation takes the form (1.38), namely $r^2-l^2=0$. In terms of the variable $\mathbf{r}$, the situation is equivalent to the one discussed in Example 1.9. Taking $f(\mathbf{r}, t)=r^2-l^2$, Eq. (1.51) reduces to $\mathbf{r} \cdot \delta \mathbf{r}=0$. Since $\mathbf{f}_2$ and $\mathbf{r}$ are collinear, there exists a scalar $\lambda$ such that $\mathbf{f}_2=\lambda \mathbf{r}$, hence $\delta W_v=\lambda \mathbf{r} \cdot \delta \mathbf{r}=0$. Inasmuch as a rigid body consists of a vast number of particles whose mutual distances are invariable, one concludes that the total virtual work done by the forces responsible for the body’s rigidity is zero.

Example 1.11 A rigid body rolls without slipping on a fixed surface. As a rule, in order to prevent slipping, a friction force between the fixed surface and the surface of the body is needed, that is, the surfaces in contact must be rough. Upon rolling without slipping, the body’s particles at each instant are rotating about an axis that contains the body’s point of contact with the surface. Thus, the friction force acts on a point of the body whose velocity at each instant is zero, because it is on the instantaneous axis of rotation. Virtual displacements are such that the body does not slip on the surface, that is, $\delta \mathbf{r}=0$ at the point of contact between the body and the fixed surface. Therefore, the virtual work done by the constraint force is $\delta W_v=\mathbf{f} \cdot \delta \mathbf{r}=0$ because $\delta \mathbf{r}=0$, even though $\mathbf{f} \neq 0$.

## 物理代写|分析力学代写Analytical Mechanics代考|Principle of Virtual Work

Newton’s formulation of mechanics is characterised by the set of differential equations
$$m_i \ddot{\mathbf{r}}_i=\mathbf{F}_i, \quad i=1, \ldots, N,$$
where $\mathbf{F}_i$ is the total or resultant force on the $i$ th particle, supposedly a known function of positions, velocities and time. This system of differential equations determines a unique solution for the $\mathbf{r}_i(t)$ once the positions and velocities are specified at an initial instant. ${ }^{10}$
In the presence of constraints, it is patently clear how inconvenient the Newtonian formulation is. First of all, it usually requires the use of more coordinates than are necessary to specify the configuration of the system. When the constraints are holonomic, for instance, the positions $\mathbf{r}_1, \ldots, \mathbf{r}_N$ are not mutually independent, making the Newtonian approach uneconomical by demanding the employment of redundant variables. Furthermore, the total force on the $i$ th particle can be decomposed as
$$\mathbf{F}_i=\mathbf{F}_i^{(a)}+\mathbf{f}_i,$$
where $\mathbf{F}_i^{(a)}$ is the applied force and $\mathbf{f}_i$ is the constraint force. In the case of the double pendulum in Example 1.4, $\mathbf{F}_1^{(a)}$ and $\mathbf{F}_2^{(a)}$ are the weights of the particles, whereas $\mathbf{f}_1$ and $\mathbf{f}_2$ are determined by the tensions on the rods or strings. The difficulty here lies in that one does not a priori know how the constraint forces depend on the positions and velocities. What one knows, in fact, are the effects produced by the constraint forces. One may also argue that the applied forces are the true causes of the motion, the constraint forces merely serving to ensure the preservation of the geometric or kinematic restrictions in the course of time. No less important is the fact that Newton’s laws – the second law together with the strong version of the third law – turn out to be incapable of correctly describing the motion of certain constrained systems (Stadler, 1982; Casey, 2014).

For all these reasons, it is highly desirable to obtain a formulation of classical mechanics as parsimonious as possible, namely involving only the applied forces and employing only independent coordinates. We shall soon see that this goal is achieved by the Lagrangian formalism when all constraints are holonomic. As an intermediate step towards Lagrange’s formulation, we shall discuss d’Alembert’s principle, which is a method of writing down the equations of motion in terms of the applied forces alone, the derivation of which explores the fact that the virtual work of the constraint forces is zero.

.

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