### 物理代写|宇宙学代写cosmology代考|Elements of relativistic point mechanics

statistics-lab™ 为您的留学生涯保驾护航 在代写宇宙学cosmology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。我们的专家在代写宇宙学cosmology代写方面经验极为丰富，各种代写宇宙学cosmology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|宇宙学代写cosmology代考|Elements of relativistic point mechanics

In the formulation of SR in terms of the Minkowski geometry, the three-dimensional quantities appearing in nonrelativistic mechanics are replaced by their four-dimensional counterparts, which leads to a more elegant, economic and transparent form of many equations. In particular, for the motion of a material point or an element of a medium, the 4-velocity is introduced
$$u^{\mu}=\frac{d x^{\mu}}{d s}=\left(\gamma, \frac{\gamma v^{i}}{c}\right), \quad \gamma \equiv \frac{1}{\sqrt{1-v^{2} / c^{2}}}$$
Here, $d s=c d t \sqrt{1-v^{2} / c^{2}}$ is an element of the interval along the particle trajectory. It is easy to verify that the vector defined in this way is normalized
$$u_{\mu} u^{\mu}=1 .$$

The 4 -acceleration of a material point or an element of a medium is
$$a^{\mu}=\frac{d u^{\mu}}{d s}=\frac{\gamma}{c} \frac{d u^{\mu}}{d t} .$$
Due to the normalization condition (2.15), the 4-velocity and 4-acceleration are mutually orthogonal: $a_{\mu} u^{\mu}=0$.

The 4-momentum of a material point (particle) is, by definition,
$$p^{\mu}=m u^{\mu}=(E / c, \vec{p}), \quad m=\text { const },$$
where $m$ is the particle rest mass, characterizing its inertial properties, $\vec{p}=\left(p^{i}\right)$ is the spatial momentum, and $E=\gamma m c^{2}$ is the particle energy in a given IRF. By definition, the squared momentum is $p^{2}=p_{\mu} p^{\mu}=m^{2} c^{2}$.

At small velocities, $v \ll c$, expanding the expression for the energy in powers of $v / c$, we obtain
$$E \approx m c^{2}+\frac{m v^{2}}{2}+O\left(v^{2} / c^{2}\right) .$$
From this relation follows a conclusion of utmost importance: that the total energy of a particle includes, in addition to the classical kinetic energy, the rest energy $m c^{2}$.

## 物理代写|宇宙学代写cosmology代考|Riemannian space–time — coordinate

So far we have been using Minkowski coordinates in Minkowski space-time, where the metric has the form (2.2), and the inertial reference frames (IRFs) connected with these coordinates. However, nothing prevents us from describing physical processes in the framework of SR with the aid of other coordinate systems, even remaining in a fixed RF in Minkowski space. For example, one can introduce spherical or cylindrical coordinates.

In the most general cases, both in Minkowski space-time and in any Riemannian space-time, and wider, in any differentiable manifold, arbitrary coordinate transformations $x^{\mu} \mapsto y^{\mu}$ are possible, with arbitrary functions
$$x^{\mu}=x^{\mu}\left(y^{0}, y^{1}, y^{2}, y^{3}\right)$$

In the physical space-time, the coordinate transformations (2.24) lead in general to changes in the reference frame.

One should note that a relationship between the notions of coordinate systems and reference frames is rather a subtle question which sometimes leads to confusions and misconceptions. It is therefore reasonable to explain in which sense we shall use these notions.
In this discussion, we will mostly follow the books $[542,578,579]$. To begin with, omitting a number of mathematical details, we can say that a Riemannian space (or space-time) is a differentiable manifold of arbitrary dimension $D$ equipped with a metric $g_{\mu \nu}$ of arbitrary signature. If it is positive-definite [the signature $(++\cdots+)]$, the space is called proper Riemannian, in other cases it is called pseudo-Riemannian, but the prefix “pseudo” is often omitted.

## 物理代写|宇宙学代写cosmology代考|Covariance, maps and atlases

Let us present the general definition of a coordinate system in a differentiable manifold. The definitions of all the corresponding notions and their detailed and rigorous discussions can be found in textbooks and monographs on global differential geometry, such as, e.g., $[285,375,461]$.

A coordinate system in a certain region $U$ of a differentiable manifold of class $k$ and dimension $D$ (or, which is the same, a map of the region $U$ ) is a one-to-one map of the region $U$ to a certain region of the arithmetic space $\mathbb{R}^{D}$. The region $U$ itself is then called the range of the map or the range of the coordinate system.

Thus each point $x$ of the region $U$ is put into correspondence with an ordered set of $D$ real numbers, which numbers are called the coordinates of this point. Moreover, if $U_{1}$ and $U_{2}$ are ranges of two maps and $x \in U_{1} \cap U_{2}$, then the coordinates of the point $x$ in one of these maps are functions of class $C^{k}$ of the coordinates of the point $x$ in the other map, with a nonzero Jacobian of the transformation. Most frequently, manifolds of the class $C^{\infty}$ are considered, with infinitely differentiable transformation functions.

The global properties of a manifold are described by sets of maps whose union of ranges covers the whole manifold (atlases). In other words, a union of ranges of maps belonging to a certain atlas is identical to the whole manifold.

Thus a coordinate system in a manifold and, in particular, in a four-dimensional Lorentzian space-time is simply a way to supply each point (event) with a certain “address” or label in the form of a set of numbers, and different ways of addressing must be related to each other by smooth transformations.

## 物理代写|宇宙学代写cosmology代考|Elements of relativistic point mechanics

pμ=米在μ=(和/C,p→),米= 常量 ,

## 物理代写|宇宙学代写cosmology代考|Riemannian space–time — coordinate

Xμ=Xμ(是0,是1,是2,是3)

## 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。