Posted on Categories:物理代写, 统计力学

# 物理代写|统计力学代写STATISTICAL MECHANICS代写|PHYS112 Classical Mechanics

avatest™

avatest.org统计力学Statistical Mechanics代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest.org， 最高质量的统计力学Statistical Mechanics作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此统计力学Statistical Mechanics作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

avatest.org™ 为您的留学生涯保驾护航 在物理Physical代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的物理Physical代写服务。我们的专家在统计力学Statistical Mechanics代写方面经验极为丰富，各种统计力学Statistical Mechanics相关的作业也就用不着 说。

## 物理代写|统计力学代写STATISTICAL MECHANICS代写|Newton’s Laws

It is obvious from the definition given here that we are dealing with an idealization, which is nevertheless approximately realized in many situations: the fact that the Earth rotates around the Sun and around itself makes a frame of reference attached to the Earth, strictly speaking, non inertial; but it can nevertheless be considered inertial for most experiments performed in laboratories.

A basic principle of mechanics is the equivalence of all inertial frames of reference, also called Galilean invariance: the laws of motion take the same form in all inertial frames of reference and the transformations between such frames consist of (constant in time) rotations and translations on a straight line at constant velocity of the origin of coordinates. This invariance implies conservation laws for total momentum, total angular momentum and energy (checking the first conservation laws will be left as exercises). ${ }^{1}$

Here we will always work in a fixed inertial frame, so we will not be concerned with Galilean invariance. Moreover, we will not discuss conservations laws apart form the conservation of energy. Newton’s first law, says, in modern terminology, that there exist inertial reference frames; since we decided to work in one such frame, we will not discuss it further.

Consider $N$ particles in $\mathbb{R}^{3}$ of masses $m_{1}, m_{2}, \ldots, m_{N}$. The position of the $i$ th particle is represented by a vector $\vec{q}{i} \in \mathbb{R}^{3}$ and the positions of all the particles of the system by a vector $\mathbf{q}=\left(\vec{q}{1}, \vec{q}{2}, \ldots, \vec{q}{N}\right) \in \mathbb{R}^{3 N}$.
Newton’s second law states that:
$$m_{i} \frac{d^{2} \vec{q}{i}}{d t^{2}}=\sum{j=1, j \neq i}^{N} \vec{F}{i j}\left(\vec{q}{i}, \vec{q}{j}\right)+\sum{i=1}^{N} \vec{F}{i}\left(\vec{q}{i}\right)$$
where $\vec{F}{i j}\left(\vec{q}{i}, \vec{q}{j}\right)$ is the force exerted on the particle of index $i$ by the one of index $j$ and $F{i}\left(\vec{q}{i}\right)$ represents the force exerted on the system by bodies located outside of it. ${ }^{2}$ We will assume that the forces are “conservative” or “derive from a potential”, namely that, for each pair $i, j$, there are smooth functions $V{i j}: \mathbb{R}^{6} \rightarrow \mathbb{R}, V_{i}: \mathbb{R}^{3} \rightarrow \mathbb{R}$, such that:
$$\vec{F}{i j}\left(\vec{q}{i}, \vec{q}{j}\right)=-\nabla{\vec{q}{i}} V{i j}\left(\vec{q}{i}, \vec{q}{j}\right)$$
$$\vec{F}{i}\left(\vec{q}{i}\right)=-\nabla_{\vec{q}{i}} V{i}\left(\vec{q}{i}\right)$$ where $\nabla{\vec{q}{i}}$ is the gradient with respect to $\vec{q}{i}$.

## 物理代写|统计力学代写STATISTICAL MECHANICS代写|Hamilton’s Equations

It is often convenient to rewrite Newton’s equations (3.2.6) in Lagrangian form or in Hamiltonian form. We will only use the latter one. ${ }^{4}$ To do so, we will introduce the phase space $\mathbb{R}^{6 N}$, and write a vector $\mathbf{x} \in \mathbb{R}^{\mathbf{6} \mathbf{N}}$ as a pair $\mathbf{x}=(\mathbf{q}, \mathbf{p})$, with $\mathbf{q}=$ $\left(\vec{q}{1}, \vec{q}{2}, \ldots, \vec{q}{N}\right) \in \mathbb{R}^{3 N}, \mathbf{p}=\left(\vec{p}{1}, \vec{p}{2}, \ldots, \vec{p}{N}\right) \in \mathbb{R}^{3 N}$.
The Hamiltonian is a function $H: \mathbb{R}^{6 N} \rightarrow \mathbb{R}$ :
$$H(\mathbf{q}, \mathbf{p})=K(\mathbf{p})+V(\mathbf{q})$$
with a kinetic energy
$$K(\mathbf{p})=\sum_{i=1}^{N} \frac{\left|\vec{p}{i}\right|^{2}}{2 m{i}}$$
and a potential energy $V(\mathbf{q})$ given by (3.2.5).
Then Hamilton’s equations are given by the following pair:
$$\frac{d \vec{q}{i}(t)}{d t}=\nabla{\vec{p}{i}} H(\mathbf{q}(t), \mathbf{p}(t)),$$ and $$\frac{d \vec{p}{i}(t)}{d t}=-\nabla_{\vec{q}{i}} H(\mathbf{q}(t), \mathbf{p}(t)),$$ for $i=1, \ldots, N$. With $H$ defined by (3.3.1), (3.3.2), these equations are: $$\frac{d \vec{q}{i}(t)}{d t}=\frac{\vec{p}{i}(t)}{m{i}}$$
and
$$\frac{d \vec{p}{i}(t)}{d t}=-\nabla{\vec{q}_{i}} V(\mathbf{q}(t)) .$$

## 物理代写|统计力学代写STATISTICAL MECHANICS代写|Newton’s Laws

$$m_{i} \frac{d^{2} \vec{q} i}{d t^{2}}=\sum j=1, j \neq i^{N} \vec{F} i j(\vec{q} i, \vec{q} j)+\sum i=1^{N} \vec{F} i(\vec{q} i)$$

$$\vec{F}{i j}(\vec{q} i, \vec{q} j)=-\nabla \vec{q} i V i j(\vec{q} i, \vec{q} j)$$ $$\vec{F} i(\vec{q} i)=-\nabla{\vec{q} i} V i(\vec{q} i)$$

## 物理代写|统计力学代写STATISTICAL MECHANICS代写|Hamilton’s Equations

$$H(\mathbf{q}, \mathbf{p})=K(\mathbf{p})+V(\mathbf{q})$$

$$K(\mathbf{p})=\sum_{i=1}^{N} \frac{|\vec{p} i|^{2}}{2 m i}$$

$$\frac{d \vec{q} i(t)}{d t}=\nabla \vec{p} i H(\mathbf{q}(t), \mathbf{p}(t)),$$

$$\frac{d \vec{p} i(t)}{d t}=-\nabla_{\vec{q} i} H(\mathbf{q}(t), \mathbf{p}(t)),$$

$$\frac{d \vec{q} i(t)}{d t}=\frac{\vec{p} i(t)}{m i}$$

$$\frac{d \vec{p} i(t)}{d t}=-\nabla \vec{q}_{i} V(\mathbf{q}(t)) .$$

## 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。