號外兒童周刋

Posted on March 29, 2019 by rpflee

Document 2_49

— 1992

2019.03.29 Friday ACHK

Finding trajectories that minimize the action

Posted on March 29, 2019 by rpflee

We have used the variational principle to determine if a given trajectory is realizable. We can also use the variational principle to find trajectories. Given a set of trajectories that are specified by a finite number of parameters, we can search the parameter space looking for the trajectory in the set that best approximates the real trajectory by finding one that minimizes the action. By choosing a good set of approximating functions we can get arbitrarily close to the real trajectory.

— Structure and Interpretation of Classical Mechanics

We have used the variational principle to determine if a given trajectory is realizable.

How?

— Me@2019-03-29 04:23:36 PM

Check if the action of that given trajectory is stationary or not.

— Me@2019-03-29 04:25:45 PM

Quantum classical logic

Posted on March 23, 2019 by rpflee

Mixed states, 2 | Eigenstates 4

— This is my guess. —

If the position is indefinite, you can express it in terms of a pure quantum state[1] (of a superposition of position eigenstates);

if the quantum state is indefinite, you can express it in terms of a mixed state;

if the mixed state is indefinite, you can express it in terms of a “mixed mixed state”[2]; etc. until definite.

At that level, you can start to use classical logic.

If you cannot get certainty, you can get certain uncertainty.

[1]: Me@2019-03-21 11:08:59 PM: This line of not correct. The uncertainty may not be quantum uncertainty. It may be classical.

[2]: Me@2019-03-22 02:56:21 PM: This concept may be useless, because a so-called “mixed mixed state” is just another mixed state.

For example, the mixture of mixed states

$\displaystyle{p |\psi_1 \rangle \langle \psi_1 | + (1-p) |\psi_2 \rangle \langle \psi_2 |}$

and

$\displaystyle{q |\phi_1 \rangle \langle \phi_1 | + (1-q) |\phi_2 \rangle \langle \phi_2 |}$

$\displaystyle{\begin{aligned} &w \bigg[ p |\psi_1 \rangle \langle \psi_1 |+ (1-p) |\psi_2 \rangle \langle \psi_2 | \bigg] + (1-w) \bigg[ q |\phi_1 \rangle \langle \phi_1 | + (1-q) |\phi_2 \rangle \langle \phi_1 | \bigg] \\ &= w p |\psi_1 \rangle \langle \psi_1 | + w (1-p) |\psi_2 \rangle \langle \psi_2 | + (1-w) q |\phi_1 \rangle \langle \phi_1 | + (1-w) (1-q) |\phi_2 \rangle \langle \phi_1 | \\ \end{aligned}}$

— This is my guess. —

— Me@2012.04.15

Find one, organize two

Posted on March 20, 2019 by rpflee

Technical debt

dna113 1 day ago [-]

I recently needed an HDMI cord for a monitor and realized that my cord drawer was accruing technical debt.

Whenever I am done with a cord I just throw it in there… it gets all tangled up with all the others. When I inevitably need one of those cords I impatiently pull it out and it makes all the other cords more tangled.

Here I am needing an HDMI cable that won’t just come out easily, I have to pay off my past laziness. But I have choices/tradeoffs/opportunities here.

I can just hurry up and get the minimum untangled and get back to watching TV.

I could untangle all of them since untangling one of them will help me untangle the others and wrap and label them.

I could just untangle the minimum, but also throw a roll of tape and a marker in there and wrap and label all future cords that go into that drawer, eventually they’ll all be nicely wrapped up and well documented.

¯\_(ツ)_/¯

jolmg 1 day ago [-]

Wow. I never thought of clutter in the home as technical debt, but it’s as similar as you describe. That really makes me see home organization in a whole new light.

— Technical Debt Is Like Tetris

— Hacker News

Whenever you have to search for something, once you have found it, organize an additional thing.

— Me@2019-03-12 11:12:28 AM

（反對）開夜車 2.4

Posted on March 19, 2019 by rpflee

本文章並(!)不(!)可作為醫學建議。如需專業意見，請諮詢專業人士。

…

至於，我自己的案例，則幸好是良性的，不會致盲；只要飛蚊不惡化，視力不會減弱。

（問：但你剛才說，飛蚊症「會帶來精神困擾」？）

無錯。可以說是心靈創傷。

（問：為什麼呢？）

如果你買了一個電腦熒光幕，發現有死點，你會怎樣做？

（問：立刻要求商戶更換。新買的電器，必定在保養期內，可以更換。）

假設不可更換呢？

（問：如果只是一兩點的話，應該不會太明顯。我會先用它，直到幾年後才再換。

但是，如果不只是一兩點的話，我可能會忍受不到；如果我負擔得起，我會立刻買過另一部。）

再假設，你永久不能更換熒光幕呢？

（問：永久有死點？）

無錯。

（問：那又真的，十分不自在。

你的意思是，「熒光幕永久有死點」就是患飛蚊症的感受？）

無錯。

所以，患了飛蚊症以後，我一聽到「不可逆轉」這四個字，就有一點兒緊張；患了飛蚊症以後，我以為，我人生不可能再快樂。

幸好，一年之後，我竟然習慣了——只要飛蚊不惡化，我就不會因為那些「死點」，而明顯不開心。

（問：「只要飛蚊不惡化」？你怎能保證？）

不能。那正正就是，我更大的苦惱。

一日不能知道，我當年開始患飛蚊症的真正原因，我也會不安，擔心症狀加劇。

「飛蚊症」只是某個或某些疾病的一個「症」，而不是「疾病」本身。一日不能知道，我的飛蚊症的病因，我也不能有力預防，症狀的加劇。

（問：你不是說，長期夜睡少睡，導致你的飛蚊症嗎？）

…

本文章並(!)不(!)可作為醫學建議。如需專業意見，請諮詢專業人士。

— Me@2019-03-18 04:47:58 PM

Physical laws are low-energy approximations to reality, 1.2

Posted on March 16, 2019 by rpflee

d_2019_01_31__23_12_26_PM_

When the temperature $\displaystyle{T}$ is higher than the critical temperature $\displaystyle{T_c}$ , point $\displaystyle{O}$ is a local minimum. So when a particle is trapped at $\displaystyle{O}$ , it is in static equilibrium.

However, when the temperature is lowered, the system changes to the lowest curve in the figure shown. As we can see, at the new state, the location $\displaystyle{O}$ is no longer a minimum. Instead, it is a maximum.

So the particle is not in static equilibrium. Instead, it is in unstable equilibrium. In other words, even if the particle is displaced just a little bit, no matter how little, it falls to a state with a lower energy.

This process can be called symmetry-breaking.

This mechanical example is an analogy for illustrating the concepts of symmetry-breaking and phase transition.

— Me@2019-03-02 04:25:23 PM

Computing Actions

Posted on March 10, 2019 by rpflee

Lagrangians in generalized coordinates

The function $\displaystyle{S_\chi}$ takes a coordinate path; the function $\displaystyle{\mathcal{S}}$ takes a configuration path.

$\displaystyle{\begin{aligned} \mathcal{S} [\gamma] (t_1, t_2) &= \int_{t_1}^{t_2} \mathcal{L} \circ \mathcal{T} [\gamma] \\ S_\chi [q] (t_1, t_2) &= \int_{t_1}^{t_2} L_\chi \circ \Gamma [q] \\ \end{aligned}}$

$\displaystyle{\begin{aligned} \mathcal{S} [\gamma] (t_1, t_2) &= S_\chi [\chi \circ \gamma] (t_1, t_2) \\ \end{aligned}}$

Computing Actions

$\displaystyle{\texttt{literal-function}}$ makes a procedure that represents a function of one argument that has no known properties other than the given symbolic name.

The method of computing the action from the coordinate representation of a Lagrangian and a coordinate path does not depend on the coordinate system.

Exercise 1.4. Lagrangian actions

For a free particle an appropriate Lagrangian is

$\displaystyle{\begin{aligned} L(t,x,v) &= \frac{1}{2} m v^2 \\ \end{aligned}}$

Suppose that $x$ is the constant-velocity straight-line path of a free particle, such that $x_a = x(t_a)$ and $x_b = x(t_b)$ . Show that the action on the solution path is

$\displaystyle{\begin{aligned} \frac{m}{2} \frac{(x_b - x_a)^2}{t_b - t_a} \\ \end{aligned}}$

— Structure and Interpretation of Classical Mechanics

$\displaystyle{\begin{aligned} L(t,x,v) &= \frac{1}{2} m v^2 \\ \end{aligned}}$

$\displaystyle{\begin{aligned} S_\chi [\gamma] (t_1, t_2) &= \int_{t_1}^{t_2} L_\chi (t, q(t), Dq(t)) dt \\ &= \int_{t_2}^{t_1} \frac{1}{2} m v^2 dt \\ &= \frac{1}{2} m v^2 \int_{t_2}^{t_1} dt \\ &= \frac{1}{2} m v^2 (t_2 - t_1) \\ &= \frac{1}{2} m (\frac{x_2 - x_1}{t_2 - t_1})^2 (t_2 - t_1) \\ &= \frac{1}{2} m \frac{(x_2 - x_1)^2}{t_2 - t_1} \\ \end{aligned}}$

— Me@2006-2008

— Me@2019-03-10 11:08:29 PM

Distinguishing hardware and software

Posted on March 10, 2019 by rpflee

X is a pattern of Y,

then

Y is hardware

and

X is software.

— Me@2019-03-10 12:10:20 AM

Time as other things, 3

Posted on March 9, 2019 by rpflee

Time travel, 3.1.3

桃花依舊, 人面全非

~ Flowers look the same as before, but the people do not.

~ Flowers go into the future.

— Me@2011.08.23

PhD, 3.4

Posted on March 7, 2019 by rpflee

碩士 4.4 | On Keeping Your Soul, 2.2.4 | Release early. Release often, 3.4

這段改編自 2010 年 4 月 18 日的對話。

…

（問：他們大概毋須，任何博士學位吧？

根本，「愛因斯坦」這名字，遠遠重要過，「博士」這銜頭。）

無錯。所以，我以上是說「理想而言」。實際而言，在你不是研究生的情況下，要「不斷做研究」，近乎沒有可能；除非，你是金庸小說中的主要角色。

（問：什麼意思？）

他們彷彿都是，既不用上班，亦毋須維生。

那樣，他們才可以全天候地，鑽研武功。

（問：那即是應該攻讀研究院？

但是，你剛才又說：

「

如果不是為了做研究，而是純粹為了穫得，碩士或博士學位的話，攻讀研究院是，十分浪費時間的。

」

你的意思是，應該為了「做研究」而讀研究院，而不應為了「得學位」？）

差不多。

這兩個目標不一定有衝突，可以並存。

但是，「做研究」應為主要目標，「得學位」應為「副作用」，或「錦上添花」。

詳細一點來說，「讀研究院」是為了獲得資源，令你有全職的時間，去研究學問。

（問：什麼「資源」？）

錢，又名「生活費」。

碩士有分兩種：「修課式碩士」和「研究式碩士」。　

要完成一個「修課式碩士」課程，你只需讀一些，深過大學本科程度的科目，考試合格就可以。你需要交學費，沒有大學資助。它適合一些工作職位上，需要碩士學歷，而毋需博士學歷的人士。換句話說，報讀「修課式碩士」的人士，通常也沒打算，繼續攻讀博士。

要完成一個「研究式碩士」課程，上課考試雖然必須，但只是次要的劇情。你「正職」要做的是，做研究。你雖然需要交學費，但大學會有資助給你。一般而言，大學給你的資助，會大於你給大學的學費。

理論上，你可以視那個差額，為你的「薪金」，或者「生活費」。有了這個「薪金」，你就可全職做研究。

（問：實際上呢？）

實際上，就當然沒有那麼理想。

— Me@2019-03-06 12:11:31 AM

Game Express

Posted on March 3, 2019 by rpflee

d_2019_01_03_15_36

2019.03.02 Saturday ACHK

Generalized Coordinates

Posted on March 1, 2019 by rpflee

Configuration Spaces

The set of all configurations of the system that can be assumed is called the configuration space of the system.

Generalized Coordinates

In order to be able to talk about specific configurations we need to have a set of parameters that label the configurations. The parameters used to specify the configuration of the system are called the generalized coordinates.
The $\displaystyle{n}$ -dimensional configuration space can be parameterized by choosing a coordinate function $\displaystyle{\chi}$ that maps elements of the configuration space to $n$ -tuples of real numbers.
The motion of the system can be described by a configuration path $\displaystyle{\gamma}$ mapping time to configuration-space points.
Corresponding to the configuration path is a coordinate path $\displaystyle{q = \chi \circ \gamma}$ mapping time to tuples of generalized coordinates.

The function $\displaystyle{\Xi \chi}$ takes the coordinate-free local tuple $\displaystyle{( t, \gamma (t), \mathcal{D} \gamma (t), ... )}$ and gives a coordinate representation as a tuple of the time, the value of the coordinate path function at that time, and the values of as many derivatives of the coordinate path function as are needed.

$\displaystyle{ \begin{aligned} \text{generalized coordinate representation} &= \Xi (\text{local tuple}) \\ (t, q(t), Dq(t), ...) &= \Xi_\chi (t, \gamma(t), \mathcal{D} \gamma(t), ...) \\ \end{aligned} }$

$\displaystyle{ \begin{aligned} \text{generalized coordinates} &= q \\ &= \chi \circ \gamma \\ \\ q(t) &= \chi(\gamma(t)) \\ \end{aligned} }$

$\displaystyle{ \begin{aligned} t &\to \gamma: \text{configuration path} \to \chi: \text{generalized coordinates} = q \\ \end{aligned} }$

$\displaystyle{ \begin{aligned} (t, q(t), Dq(t), ...) &= \Xi_\chi (t, \gamma(t), \mathcal{D} \gamma(t), ...) \\ \\ \Gamma[q](t) &= (t, q(t), Dq(t), ...) \\ \Gamma[q] &= \Xi_\chi \circ \mathcal{T}[\gamma] \\ \end{aligned} }$

— 1.2 Configuration Spaces

— Structure and Interpretation of Classical Mechanics

— Me@2019-03-01 03:09:25 PM

2019.03.01 Friday ACHK

Physics Town

continues

Monthly Archives: March 2019

號外兒童周刋

Finding trajectories that minimize the action

Quantum classical logic

Find one, organize two

（反對）開夜車 2.4

Physical laws are low-energy approximations to reality, 1.2

Computing Actions

Lagrangians in generalized coordinates

Computing Actions

Exercise 1.4. Lagrangian actions

Distinguishing hardware and software

Time as other things, 3

PhD, 3.4

Game Express

Generalized Coordinates

Configuration Spaces

Generalized Coordinates