# 2007, 3

— Me@2021-04-19 08:39:47 PM

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# Ex 1.21 The dumbbell, 3

Structure and Interpretation of Classical Mechanics

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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.

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[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}}

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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}}

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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}}

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\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

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# Causal diamonds in time travel

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

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

— Me@2021-04-17 03:12:58 PM

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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

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# 黎明 3

— 黎明

— Me@2016-05-16 05:58:16 PM

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2021.04.17 Saturday ACHK

# To realize is to realize, 1.2

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.

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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

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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.

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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.

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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

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# PC Home

— Me@2021-04-13 05:40:22 PM

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# 2.9 Lightlike compactification, d

A First Course in String Theory

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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.

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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

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

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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

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# Time travel, 3.2

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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

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# 龍珠 3

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

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# 相聚零刻

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（我只會因自身的問題，而頹廢怠慢，無所事事。）

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「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。

— Me@2021-04-11 07:10:08 AM

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# 2004, 3

— Me@2021-04-10 05:06:41 PM

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# Ex 1.21 The dumbbell, 2

Structure and Interpretation of Classical Mechanics

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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):

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

\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] \\ \end{aligned}}

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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}}

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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

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# Universal wave function, 21

For all, 9

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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

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universe ~ 100%

But 100% of what?

— Me@2021-04-09 05:20:23 PM

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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

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# Privacy 2.2

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

— Me@2017-02-01 12:21:24 PM

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# 一萬個小時 3.2

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（安：但是我的工作，幾乎沒有一些方面，需要任何的學術成份。）

— Me@2021-04-08 02:11:07 AM

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# MSI GTX 1050 TI GAMING X 4G

— Me@2021-04-07 04:42:48 PM

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# 2.9 Lightlike compactification, c

A First Course in String Theory

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

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\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} }

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\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

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# Being conscious

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What is the meaning of “a company exists”?

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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.

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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

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# Story

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

— Me@2011.08.27

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# 光速飛行 1.2

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（我只會因自身的問題，而頹廢怠慢，無所事事。）

— Me@2021-04-05 06:14:46 PM

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