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## 物理代写|粒子物理代写Particle Physics代考|Elements of Classical Field Theory

Although the natural framework to describe the interactions among elementary particles is the quantum theory, we will start here by recalling some elements of the classical theory of fields. There is a good reason for that. As we alluded to in the first chapter, in order to obtain a quantum theory we start from the corresponding classical theory to which we apply the quantisation prescription. This applies to any physical system, no matter whether it has a finite or an infinite number of degrees of freedom. It follows that the knowledge of the classical system is essential in the formulation of the corresponding quantum system. Since the quantum theory of fields will be the language of elementary particle physics, it is essential to understand the corresponding classical field theory.

## 物理代写|粒子物理代写Particle Physics代考|Lagrangian and Hamiltonian Mechanics

The high level of conceptualisation of classical mechanics has, since the 19 th century, played an essential part in the development of physical theories. We shall give here a very brief review of the main results with no proofs, essentially in order to fix terminology and notations.

The Lagrangian. Let us consider a system with $N$ degrees of freedom and let $q_{a}(t), a=1, \ldots, N$, denote the corresponding generalised coordinates. We will assume that they determine a point $\boldsymbol{q}$ in an $N$-dimensional differentiable manifold $\mathcal{M}$, for example the $N$-dimensional real space $\mathbb{R}^{N} .{ }^{1}$ We shall call $\mathcal{M}$ the configuration space of the system. Since $\mathcal{M}$ is differentiable, we can consider at every point $\boldsymbol{q}$ the set of $N$ tangent vectors $\dot{q}{a}(t)=\mathrm{d} q{a}(t) / \mathrm{d} t$ of curves passing through $\boldsymbol{q}$. Together with $q_{a}$ they span a $2 N$-dimensional space, which we shall call $\mathcal{T}(\mathcal{M}) .{ }^{2}$

A Lagrangian $L$ is a real function of the $2 N$ variables $q_{a}$ and $\dot{q}{a}$ and, possibly, the time $t$, i.e. $L\left(q{a}, \dot{q}_{a}, t\right): \mathcal{T}(\mathcal{M}) \times \mathbb{R} \rightarrow \mathbb{R}$. An important mathematical tool, which was developed for functional analysis problems, of the kind we shall deal with in this book, is the calculus of variations. For the simple case of $\mathcal{M}=\mathbb{R}^{N}$ it derives the following well-known theorem:

• Consider $\boldsymbol{q} \in \mathbb{R}^{N}$ and let $\gamma=\left{t, \boldsymbol{q} \mid \boldsymbol{q}=\boldsymbol{q}(t), t_{0} \leq t \leq t_{1}\right}$ be a curve in $\mathbb{R}^{N} \times \mathbb{R}$ such that $\boldsymbol{q}\left(t_{0}\right)=\boldsymbol{q}{0}$ and $\boldsymbol{q}\left(t{1}\right)=\boldsymbol{q}{1}$, and let the Lagrangian $L: \mathbb{R}^{N} \times \mathbb{R}^{N} \times \mathbb{R} \rightarrow \mathbb{R}$ be a sufficiently regular function of $2 N+1$ variables. We can prove that the curve $\gamma$ is extremal for the action functional defined by $S[\gamma]=\int{t_{0}}^{t_{1}} L(\boldsymbol{q}, \dot{\boldsymbol{q}}, t) \mathrm{d} t$ in the space of the curves joining $\left(t_{0}, \boldsymbol{q}{0}\right)$ to $\left(t{1}, \boldsymbol{q}_{1}\right)$ if and only if the Euler-Lagrange equations
$$\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{\partial L}{\partial \dot{q}}\right)-\frac{\partial L}{\partial \boldsymbol{q}}=0$$
are satisfied along $\gamma$.

## 物理代写|粒子物理代写Particle Physics代考|Lagrangian and Hamiltonian Mechanics

$$\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{\partial L}{\partial \dot{q}}\right)-\frac{\partial L}{\partial \boldsymbol{q}}=0$$

## 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。