物理代写|PHY5246 Theoretical Dynamics
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PHY5246 Theoretical Dynamics课程简介
Classical mechanics studies the motion of material bodies. It is among the fundamental branches of modern physics and is therefore an essential component of all graduate programs in Physics. This course introduces the structure of classical mechanics and discusses some of its most important applications in modern physics.
In the first part of this course, we will start from the Lagrangian formulation of classical mechanics and study the solution of the equations of motion of several systems, from simple one-body systems, to more complex systems acted upon by central forces and rigid bodies, up to scattering problems and oscillations. We will also extend our discussion to include the theory of special relativity.
The second part of this course will introduce the hamiltonian formulation of classical mechanics and study its formal and physical consequences. More advanced topics, such as non-linear dynamics and continuous systems, will be discussed in the third part of this course, depending on time availability.
PREREQUISITES
Classical mechanics studies the motion of material bodies. It is among the fundamental branches of modern physics and is therefore an essential component of all graduate programs in Physics. This course introduces the structure of classical mechanics and discusses some of its most important applications in modern physics.
In the first part of this course, we will start from the Lagrangian formulation of classical mechanics and study the solution of the equations of motion of several systems, from simple one-body systems, to more complex systems acted upon by central forces and rigid bodies, up to scattering problems and oscillations. We will also extend our discussion to include the theory of special relativity.
The second part of this course will introduce the hamiltonian formulation of classical mechanics and study its formal and physical consequences. More advanced topics, such as non-linear dynamics and continuous systems, will be discussed in the third part of this course, depending on time availability.
PHY5246 Theoretical Dynamics HELP（EXAM HELP， ONLINE TUTOR）
Theorem 1.6.2 $A^{-1}$ exists only when $\operatorname{det} A \neq 0$. The elements are then found by:
$$
\left(a^{-1}\right){i j}=\frac{U{j i}}{\operatorname{det} A} .
$$
(Note the order of the indexes!)
Proof Let $\widehat{A}=\left(\alpha_{i j}=U_{j i}\right)$ be an $(n \times n)$-matrix. With the expansion theorems (1.327) and (1.332) we find:
$$
\begin{aligned}
& \operatorname{det} A=\sum_j a_{i j} U_{i j}=\sum_j a_{i j} \alpha_{j i}=(A \cdot \widehat{A}){i i}, \ & \operatorname{det} A=\sum_i a{i j} U_{i j}=\sum_i \alpha_{j i} a_{i j}=(\widehat{A} \cdot A){j j} . \ & \end{aligned} $$ The diagonal elements of the product matrices $A \cdot \widehat{A}$ and $\widehat{A} \cdot A$ are thus all identical to $\operatorname{det} A$. What about the non-diagonal elements? With (1.336) one finds: $$ (A \cdot \widehat{A}){i j}=\sum_k a_{i k} \alpha_{k j}=\sum_k a_{i k} U_{j k}=0 \quad \text { for } i \neq j .
$$
It follows that $A \cdot \widehat{A}$ and $\widehat{A} \cdot A$ are diagonal matrices with
$$
A \cdot \widehat{A}=\widehat{A} \cdot A=\operatorname{det} A \cdot \mathbb{1}
$$
With $\operatorname{det} A \neq 0$ and by comparison with (1.337) the theorem is proved:
$$
\frac{\widehat{A}}{\operatorname{det} A}=A^{-1} \Longleftrightarrow \frac{U_{j i}}{\operatorname{det} A}=\left(a^{-1}\right)_{i j} .
$$
By the term function $f(x)$ one understands the unique attribution of a dependent variable $y$ from the co-domain $W$ to an independent variable $x$ from the domain of definition $D$ of the function $f(x)$ :
$$
y=f(x) \quad ; \quad D \subset \mathbb{R} \stackrel{f}{\longrightarrow} W \subset \mathbb{R} .
$$
We ask ourselves how $f(x)$ changes with $x$. All elements of the sequence
$$
\left{x_n\right}=x_1, x_2, x_3, \cdots, x_n, \cdots
$$ shall be from the domain of definition of the function $f$. Then for each $x_n$ there exists a
$$
y_n=f\left(x_n\right)
$$
and therewith a ‘new’ sequence $\left{f\left(x_n\right)\right}$.
Definition $f(x)$ possesses at $x_0$ a limiting value $f_0$, if for each sequence $\left{x_n\right} \rightarrow x_0$ holds:
$$
\lim {n \rightarrow \infty} f\left(x_n\right)=f_0 $$ That is written as: $$ \lim {x \rightarrow x_0} f(x)=f_0
$$
Examples
1.
$$
f(x)=\frac{x^3}{x^3+x-1} \quad ; \quad \lim {x \rightarrow \infty} f(x)=? $$ This expression can be reformulated for all $x \neq 0$ : $$ f(x)=\frac{1}{1+\frac{1}{x^2}-\frac{1}{x^3}} . $$ For all sequences $\left{x_n\right}$, which tend to $\infty, \frac{1}{x^2}$ and $\frac{1}{x^3}$ become null sequences. That means: $$ \lim {x \rightarrow \infty} \frac{x^3}{x^3+x-1}=1
$$
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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