2007, 3

Posted on April 19, 2021 by rpflee

— Me@2021-04-19 08:39:47 PM

Ex 1.21 The dumbbell, 3

Posted on April 18, 2021 by rpflee

Structure and Interpretation of Classical Mechanics

c. Make a change of coordinates to a coordinate system with center of mass coordinates $x_{cm}$ , $y_{cm}$ , angle $\theta$ , distance between the particles $c$ , and tension force $F$ . Write the Lagrangian in these coordinates, and write the Lagrange equations.

~~~

[guess]

$\displaystyle{ \begin{aligned} y_{cm} &= \frac{m_0 y_0 + m_1 y_1}{m_0 + m_1} \\ x_{cm} &= \frac{m_0 x_0 + m_1 x_1}{m_0 + m_1} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} y_{cm} &= \frac{m_0 y_0 + m_1 (y_0 + c(t) \sin \theta)}{m_0 + m_1} \\ x_{cm} &= \frac{m_0 x_0 + m_1 (x_0 + c(t) \cos \theta)}{m_0 + m_1} \\ \end{aligned}}$

Let $\displaystyle{M = m_0 + m_1}$ .

$\displaystyle{ \begin{aligned} y_{cm} &= \frac{M y_0 + m_1 c(t) \sin \theta}{M} \\ x_{cm} &= \frac{M x_0 + m_1 c(t) \cos \theta}{M} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} y_0 &= y_{cm} - \frac{m_1}{M} c(t) \sin \theta \\ x_0 &= x_{cm} - \frac{m_1}{M} c(t) \cos \theta \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} y_1 &= y_{cm} + \frac{m_0}{M} c(t) \sin \theta \\ x_1 &= x_{cm} + \frac{m_0}{M} c(t) \cos \theta \\ \end{aligned}}$

The Lagrangian

$\displaystyle{ \begin{aligned} L &= \frac{1}{2} m_0 (\dot x_0^2 + \dot y_0^2) + \frac{1}{2} m_1 (\dot x_1^2 + \dot y_1^2) + \lambda \left[ (x_1(t) - x_0(t))^2 + (y_1(t) - y_0(t))^2 - l^2 \right] \\ &= \frac{1}{2} m_0 (\dot x_0^2 + \dot y_0^2) + \frac{1}{2} m_1 (\dot x_1^2 + \dot y_1^2) - \frac{F}{2l} \left[ (c(t))^2 - l^2 \right] \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \dot y_0 &= \dot y_{cm} - \frac{m_1}{M} \dot c(t) \sin \theta - \frac{m_1}{M} c(t) \dot \theta \cos \theta\\ \dot x_0 &= \dot x_{cm} - \frac{m_1}{M} \dot c(t) \cos \theta - \frac{m_1}{M} c(t) \dot \theta \sin \theta \\ \end{aligned}}$

…

[guess]

— Me@2021-04-17 05:40:46 PM

Causal diamonds in time travel

Posted on April 17, 2021 by rpflee

Quantum mechanics is a set of rules that allows an observer to predict, explain, and/or verify observations (and especially their mutual relationships) that he has access to.

…

No observer can detect inconsistencies within the causal diamonds. However, inconsistencies between “stories” as told by different observers with different causal diamonds are allowed (and mildly encouraged) in general (as long as there is no observer who could incorporate all the data needed to see an inconsistency).

— Raphael Bousso is right about firewalls

— Lubos Motl

There is no “god’s eye view” in physics.

— Me@2021-04-17 03:12:58 PM

Macroscopic time is with respect to an observer. Actually, physics is with respect to an observer.

In the real universe, any observer’s observations must be consistent. When two observers compare their observations, their results must be consistent, because the comparison itself is an observation of an observer.

Time travel in the absolute sense is logically impossible. Let’s assume that it is logically possible.

If a time travel story follows the principle of “an observer’s observations must be consistent”, each character in that story must see a consistent timeline, even if different characters’ timelines may be inconsistent. That is fine as long as such inconsistent observers never meet to compare their results.

If two of such observers choose to meet to compare their results, the action to “meet to compare” itself will render the results consistent. It is similar to the resolution of the twin paradox in special relativity.

There is no “god’s eye view” in physics. Every physical event must be described with respect to an observer. Every physical event, even if the event is “to compare observation results”, must be described with respect to an observer.

— Me@2017-05-10 07:45:36 AM

黎明 3

Posted on April 17, 2021 by rpflee

講到做唔到，唔好講。

好辛苦做咗啲嘢無人知，唔好做。

— 黎明

— Me@2016-05-16 05:58:16 PM

2021.04.17 Saturday ACHK

To realize is to realize, 1.2

Posted on April 16, 2021 by rpflee

For example, when you ask “how to be a polite person”, you become a polite person.

In other words, when you realize that you should be a polite person, that polite person is realized at that moment.

The ultimate self-fulfilling prophecies:

1. free will or not

2. god or no god

3. afterlife or not

4. future spouse exists or not

Why self-fulfilling?

1. free will or not

If we knew the exact physical laws and all the data of the present state of the universe, we would be able to predict, with 100% accuracy, the state of the universe at any future moment.

In theory, the exact physical laws exist, whether we know them or not. Also, in theory, all the details of the present state of the universe exist, whether we know them or not.

So in theory, there is no free will, because the future is already fixed, by the physical laws.

However, even if we knew the exact physical laws, it would be still logically impossible to get all the data of the present state of the whole universe, because it is logically impossible for any observer to observe itself directly. For example, no camera can take a picture of itself directly.

So “in practice”, which is actually “in theory”, there is free will, because logically, no one can predict, with 100% accuracy, your future actions.

— Me@2021-04-16 04:45:47 PM

PC Home

Posted on April 14, 2021 by rpflee

— Me@2021-04-13 05:40:22 PM

2.9 Lightlike compactification, d

Posted on April 14, 2021 by rpflee

A First Course in String Theory

Represent your answer to part (c) in a spacetime diagram. Show two points related by the identification (2) and the space and time axes for the Lorentz frame $S'$ in which the compactification is standard.

~~~

Note 1:

The identifications $\displaystyle{ x \sim x + 0 }$ and $\displaystyle{ x \sim x + \infty }$ have the same meaning, which is that $x$ has no identification at all. In other words, there is NO non-zero real number $r$ such that

$\displaystyle{ x \sim x + r }$

Note 2:

$\displaystyle{ \begin{aligned} \begin{bmatrix} ct \\ x \\ \end{bmatrix} &\sim \begin{bmatrix} ct - 2 \pi R \\ x + 2 \pi \sqrt{R^2 + R_S^2} \\ \end{bmatrix} \end{aligned} }$

This identification must be done to both the space and time coordinates. In other words, it cannot be done to only one of $ct$ and $x$ .

— Me@2021-04-13 12:02:13 PM

Time travel, 3.2

Posted on April 13, 2021 by rpflee

時光機 3.2 | Time is other things, 4 | Cumulative concept of time, 9.2

If you keep deleting your blog posts one by one, your blog effectively keeps becoming its past. However, its environment is still cumulative. In other words, the environment is not becoming its past.

If the environment and the blog interchange labels, the blog’s environment is going to the past, while the blog is not.

— Me@2013-08-10 07:41 PM

— Me@2021-04-12 06:02 PM

龍珠 3

Posted on April 12, 2021 by rpflee

On knowledge independent vectors:

When you already have a complete set of vectors in $n$ dimensional space, you have to go up one extra dimension in order to find a new independent vector.

— Me@2011.08.27

相聚零刻

Posted on April 11, 2021 by rpflee

光速飛行 1.2.2 | 尋覓 2.2.3.2

我是一個奇怪的人。我比一般人益智和勵志。我不會因為愛情問題，而無心向學，放慢工作。我既不會因為單身一人，而茶飯不思，亦不會因為有心上人，而心神恍惚。

（我只會因自身的問題，而頹廢怠慢，無所事事。）

大一時，在某一活動中遇到她。之後，我該組還有一次後續的活動聚會。但是，她剛巧沒有參加。所以，之後就再沒有機會見到她。我和她僅有的相處，就只是於 ICQ（一款網絡即時通訊軟件）中的聊天，為數不多。

後來，一年半載後的後來，我就在和她不太熟絡（其實是不太認識）的情況下，向她表白。那十分突然，被拒絶是情理之中，意料之內。但是，我當時也沒有辦法，因為就快畢業，再不說就沒有機會說了。

我當時的主要目的是，以防一個情況是，原來她也暗戀我而又不敢說，而導致雙方錯失良緣。然後，又要輪迴多少世，才再相遇，十分可惜。

其實，以我現在的智力，是有辦法的。但是，我當時就只有，當時的智力。以當時的智力，我實在沒有辦法。

之後，我又不會時常想到她。但是，每當想起她時，仍然是會有快樂的情緒。

「拒絶」是她的權利，而且是正確的決定。只要是她的決定，就是正確的決定。所以其實，她是沒有做錯任何事。她在品德上沒有問題，所以，我仍然會是十分欣賞她的。我的人生之中，可以遇過如此仙子，已經是莫大的榮幸。

只不過，我不會再對她，有愛情上的喜歡；因為喜歡人的先決條件是，你感到對方也喜歡你，或有機會喜歡你。愛情是雙向的，單向的不是愛情。心上人已拒絶，那就證明，她不是我的未來人，我自然不會再花，太多的心神時間去想念她。

換句話說，我不會在「肯定對方對我沒有愛情感覺」的情況下，仍然對她，有愛情的感覺。在愛情上，我只會愛自己的（未來）太太。

— Me@2021-04-11 07:10:08 AM

2004, 3

Posted on April 10, 2021 by rpflee

— Me@2021-04-10 05:06:41 PM

Ex 1.21 The dumbbell, 2

Posted on April 10, 2021 by rpflee

Structure and Interpretation of Classical Mechanics

Notice that these forces of constraint are proportional to the normal to the constraint surface at each instant and thus do no work for motions that obey the constraint.

b. Write the formal Lagrangian

$\displaystyle{L(t;x_0, y_0, x_1, y_1, F; \dot x_0, \dot y_0, \dot x_1, \dot y_1, \dot F)}$

such that Lagrange’s equations will yield the Newton’s equations you derived in part a.

~~~

[guess]

Eq (1.99):

$\displaystyle{L(t;x,\lambda; \dot x, \dot \lambda) = \sum_\alpha \frac{1}{2} m_\alpha \dot{\mathbf{x}}_\alpha^2 - V(t,x) + \lambda \phi(t,x)}$

The Lagrange equation:

$\displaystyle{ \begin{aligned} D ( \partial_2 L \circ \Gamma[q]) - (\partial_1 L \circ \Gamma[q]) &= 0 \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \frac{d}{dt} \frac{\partial L}{\partial \dot x_0} - \frac{\partial L}{\partial x_0} &= 0 \\ \frac{d}{dt} \frac{\partial L}{\partial \dot x_1} - \frac{\partial L}{\partial x_1} &= 0 \\ \frac{d}{dt} \frac{\partial L}{\partial \dot y_0} - \frac{\partial L}{\partial y_0} &= 0 \\ \frac{d}{dt} \frac{\partial L}{\partial \dot y_1} - \frac{\partial L}{\partial y_1} &= 0 \\ \frac{d}{dt} \frac{\partial L}{\partial \dot \lambda} - \frac{\partial L}{\partial \lambda} &= 0 \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \frac{d}{dt} \left( m_0 \dot x_0 \right) + \left( \lambda 2 (x_1 - x_0)\right) &= 0 \\ \frac{d}{dt} \left( m_0 \dot x_1 \right) - \left( \lambda 2 (x_1 - x_0)\right) &= 0 \\ \frac{d}{dt} \left( m_0 \dot y_0 \right) + \left( \lambda 2 (y_1 - y_0)\right) &= 0 \\ \frac{d}{dt} \left( m_0 \dot y_1 \right) - \left( \lambda 2 (y_1 - y_0)\right) &= 0 \\ \frac{d}{dt} \left( 0 \right) - \left[ (x_1(t) - x_0(t))^2 + (y_1(t) - y_0(t))^2 - l^2 \right] &= 0 \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} m_0 \ddot x_0 &= - 2 \lambda (x_1 - x_0) \\ m_0 \ddot x_1 &= 2 \lambda (x_1 - x_0) \\ m_0 \ddot y_0 &= - 2 \lambda (y_1 - y_0) \\ m_0 \ddot y_1 &= 2 \lambda (y_1 - y_0) \\ (x_1(t) - x_0(t))^2 + (y_1(t) - y_0(t))^2 - l^2 &= 0 \\ \end{aligned}}$

Let $\displaystyle{\lambda = - \frac{F}{2l}}$ :

$\displaystyle{ \begin{aligned} m_0 \ddot x_0 &= F \frac{(x_1 - x_0)}{l} \\ m_0 \ddot x_1 &= -F \frac{(x_1 - x_0)}{l} \\ m_0 \ddot y_0 &= F \frac{(y_1 - y_0)}{l} \\ m_0 \ddot y_1 &= -F \frac{(y_1 - y_0)}{l} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} m_0 \ddot y_0 &= F \sin \theta \\ m_0 \ddot x_0 &= F \cos \theta \\ m_0 \ddot y_1 &= -F \sin \theta \\ m_0 \ddot x_1 &= -F \cos \theta \\ \end{aligned}}$

[guess]

— Me@2021-04-09 11:18:07 PM

Universal wave function, 21

Posted on April 9, 2021 by rpflee

For all, 9

A problem of universal wave function (universe) is that universe is a relative concept.

Another problem is that wave function is also.

— Me@2017-05-10 05:46:44 PM

— Me@2021-04-09 06:25:07 PM

universe ~ 100%

But 100% of what?

— Me@2021-04-09 05:20:23 PM

The wave function is expressed in terms of basis state vectors.

So it will have a different form if you choose a different basis.

— Me@2021-04-09 06:29:20 PM

Privacy 2.2

Posted on April 9, 2021 by rpflee

Privacy means the right to separate your private and public identities.

— Me@2017-02-01 12:21:24 PM

一萬個小時 3.2

Posted on April 9, 2021 by rpflee

這段改編自 2010 年 4 月 24 日的對話。

（安：但是我的工作，幾乎沒有一些方面，需要任何的學術成份。）

或者，未必一定是一些，十分高層次的學術課題。在你正職當中，乘機磨鍊一些才藝，都可以。例如，我在之前的工作，訓練了表達技巧。我在中學教書時，需要每天講學三個多小時。講了一年半載後，說話才通順，可以完成完整句子而不「跳線」。

那個經驗，令我知道任何學問，或者才藝，如果要訓練自己，純熟到足以用來維生賺錢，或者娛樂滿足，需要的時間幅度是，一日幾小時，持續一兩年，才會開竅；然後，再持續一兩年，才會知道自己，在該門學問才藝，為何會開竅，開了什麼竅；又再過一兩年，才會有能力教人，如何在該門學問才藝開竅。

我現在的狀態是，在某些範疇中，有能力教人之餘，還有能力教人如何教。亦即是話，如果教員甲的教學質素，好過教員乙的話，我可以解釋到，甲比乙好在哪裡。

— Me@2021-04-08 02:11:07 AM

MSI GTX 1050 TI GAMING X 4G

Posted on April 7, 2021 by rpflee

— Me@2021-04-07 04:42:48 PM

2.9 Lightlike compactification, c

Posted on April 7, 2021 by rpflee

A First Course in String Theory

… Find the velocity parameter of $\displaystyle{S'}$ with respect to $\displaystyle{S}$ and the compactification radius in the Lorentz frame $\displaystyle{S'}$ .

~~~

$\displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ x' \end{bmatrix} &= \begin{bmatrix} \gamma & -\beta \gamma \\ -\beta \gamma & \gamma \\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \end{bmatrix} \\ \end{aligned} }$

$\displaystyle{ \begin{aligned} \begin{bmatrix} ct \\ x \\ \end{bmatrix} &\sim \begin{bmatrix} ct - 2 \pi R \\ x + 2 \pi \sqrt{R^2 + R_S^2} \\ \end{bmatrix} \end{aligned} }$

$\displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ x' \end{bmatrix} &\sim \begin{bmatrix} ct' \\ x' \\ \end{bmatrix} + \begin{bmatrix} \gamma & -\beta \gamma \\ -\beta \gamma & \gamma \\ \end{bmatrix} \begin{bmatrix} - 2 \pi R \\ 2 \pi \sqrt{R^2 + R_S^2} \\ \end{bmatrix} \\ \end{aligned} }$

$\displaystyle{ \begin{aligned} R + \beta \sqrt{R^2 + R_S^2} &= 0 \\ \beta^2 &= \frac{R^2}{R^2 + R_S^2} \\ \gamma &= \sqrt{\frac{R^2 + R_S^2}{R_S^2}} \\ \end{aligned} }$

$\displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ x' \end{bmatrix} &\sim \begin{bmatrix} ct' \\ x' \\ \end{bmatrix} + \begin{bmatrix} 0 \\ R_S \\ \end{bmatrix} 2 \pi \\ \end{aligned} }$

— Me@2021-04-07 07:02:14 AM

Being conscious

Posted on April 6, 2021 by rpflee

清醒

What is the meaning of “a company exists”?

A company exists when the people inside it cooperate.

They have a leader, getting the needed information by using the employees as sources.

If a company’s employees no longer cooperate or its leader can no longer get the information needed, that company no longer exists.

What is the meaning of “being conscious”?

A brain is conscious if its different (hardware or software) parts cooperate by exchanging information.

— Me@2013-08-10 06:48 PM

— Me@2021-04-06 10:44 PM

Story

Posted on April 6, 2021 by rpflee

A story means data with a context. A storyline is a linear and more understandable context.

— Me@2011.08.27

光速飛行 1.2

Posted on April 6, 2021 by rpflee

姻緣伏線 1.3 | 尋覓 2.2.3

這段改編自 2010 年 10 月 14 日的對話。

而方法就是，不斷地去見人。間中相約朋友，舊朋友自然會，介紹新朋友。參加自己有興趣的活動，例如去學日文，你自然會遇到一堆，志同道合的朋友。

同時，你亦要不斷地去，增值自己，令到自己是，對方的可能對象中，最好的一個。

我是一個奇怪的人。我比一般人益智和勵志。我不會因為愛情問題，而無心向學，放慢工作。我既不會因為單身一人，而茶飯不思，亦不會因為有心上人，而心神恍惚。

（我只會因自身的問題，而頹廢怠慢，無所事事。）

我於大學時代，頗為積極參加活動。積極參加活動的嚴重後果是，會在其中一個活動中，遇到喜歡的人。

之後，自然會間中思念她，想著如何找她傾談，如何再遇到她。

但是，「思念」不單不會，傷害我的學業，反而令我多了，一個主要理由，去發奮向上；因為，如果我他日，追求成功的話，她就有很大機會是，我的未來太太；我就有一個絕對的責任，給她提供一個，安穩的經濟環境。

如果不是學業有功，和事業有成，達到這目標的機會，就會低很多。

— Me@2021-04-05 06:14:46 PM

Physics Town

continues

Author Archives: rpflee

2007, 3

Ex 1.21 The dumbbell, 3

Causal diamonds in time travel

黎明 3

To realize is to realize, 1.2

PC Home

2.9 Lightlike compactification, d

Time travel, 3.2

龍珠 3

相聚零刻

2004, 3

Ex 1.21 The dumbbell, 2

Universal wave function, 21

Privacy 2.2

一萬個小時 3.2

MSI GTX 1050 TI GAMING X 4G

2.9 Lightlike compactification, c

Being conscious

Story

光速飛行 1.2