# Logical arrow of time, 10

Two distinguishable macrostates can both evolve into one indistinguishable macrostate.

— Me@2013-08-11 11:08 AM

.

.

# Ultimate Freedom, 2

Imagination is not an empirical or superadded power of consciousness, it is the whole of consciousness as it realizes its freedom.

— Jean-Paul Sartre

.

It is where anything not forbidden can happen.

It is where anything logically consistent can happen.

— Me@2016-11-20 09:14:06 AM

— Me@2021-09-26 09:22:54 PM

.

.

# 改變因果鏈起點

.

（安：我覺得你對我最大的價值，反而是：

.

.

— Me@2021-09-22 10:24:46 PM

.

.

# 1990s, 5.2

— Photomyne colorization

— Me@2021-09-19 06:13:39 PM

.

.

# Ex 1.21 The dumbbell, 3.3

Structure and Interpretation of Classical Mechanics

.

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

~~~

[guess]


(define (KE-particle m v)
(* 1/2 m (square v)))

(define ((L-free-constrained m0 m1 l) local)
(let* ((extract (extract-particle 2))
(p0 (extract local 0))
(q0 (coordinate p0))
(qdot0 (velocity p0))

(p1 (extract local 1))
(q1 (coordinate p1))
(qdot1 (velocity p1))

(F (ref (coordinate local) 4)))

(- (+ (KE-particle m0 qdot0)
(KE-particle m1 qdot1))
(U-constraint q0 q1 F l))))

(define ((extract-particle pieces) local i)
(let* ((indices (apply up (iota pieces (* i pieces))))
(extract (lambda (tuple)
(vector-map (lambda (i)
(ref tuple i))
indices))))
(up (time local)
(extract (coordinate local))
(extract (velocity local)))))

(define (U-constraint q0 q1 F l)
(* (/ F (* 2 l))
(- (square (- q1 q0))
(square l))))

(let ((L (L-free-constrained 'm_0 'm_1 'l))
(q-rect (up (literal-function 'x_0)
(literal-function 'y_0)
(literal-function 'x_1)
(literal-function 'y_1)
(literal-function 'F))))
(show-expression
((compose L (Gamma q-rect)) 't))) $\displaystyle{ \frac{1}{2} m_0 \left( \dot x_0^2 + \dot y_0^2 \right) + \frac{1}{2} m_1 \left( \dot x_1^2 + \dot y_1^2 \right) + \frac{F}{2 l} \left( l^2 - y_1^2 + 2 y_0 y_1 - x_1^2 + 2 x_0 x_1 - y_0^2 - x_0^2 \right) }$ $\displaystyle{ = \frac{1}{2} m_0 \left( \dot x_0^2 + \dot y_0^2 \right) + \frac{1}{2} m_1 \left( \dot x_1^2 + \dot y_1^2 \right) - \frac{F}{2 l} \left[ (y_1 - y_0)^2 + (x_1 - x_0)^2 - l^2 \right] }$


(define ((q->r m0 m1) local)
(let ((q (coordinate local)))
(let ((x_cm (ref q 0))
(y_cm (ref q 1))
(theta (ref q 2))
(c (ref q 3))
(F (ref q 4))
(M (+ m0 m1)))
(let ((x0 (- x_cm (* (/ m1 M) c (cos theta))))
(y0 (- y_cm (* (/ m1 M) c (sin theta))))
(x1 (+ x_cm (* (/ m0 M) c (cos theta))))
(y1 (+ y_cm (* (/ m0 M) c (sin theta)))))
(up x0 y0 x1 y1 F)))))

(let ((q (up (literal-function 'x_cm)
(literal-function 'y_cm)
(literal-function 'theta)
(literal-function 'c)
(literal-function 'F))))
(show-expression (q 't))) (show-expression
(up 't
(up 'x_cm 'y_cm 'theta 'c 'F)
(up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot))) (show-expression
((q->r 'm_0 'm_1)
(up 't
(up 'x_cm 'y_cm 'theta 'c 'F)
(up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot)))) (let ((q (up (literal-function 'x_cm)
(literal-function 'y_cm)
(literal-function 'theta)
(literal-function 'c)
(literal-function 'F))))
(show-expression ((q->r 'm_0 'm_1) ((Gamma q) 't)))) (show-expression
((F->C (q->r 'm_0 'm_1))
(up 't
(up 'x_cm 'y_cm 'theta 'c 'F)
(up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot)))) (define (L-cm m0 m1 l)
(compose
(L-free-constrained m0 m1 l) (F->C (q->r m0 m1))))

(show-expression
((L-cm 'm_0 'm_1 'l)
(up 't
(up 'x_cm 'y_cm 'theta 'c 'F)
(up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot))))  $\displaystyle{ \frac{1}{\mu} = \frac{1}{m_0} + \frac{1}{m_1} }$ $\displaystyle{ L_{cm} }$ $\displaystyle{ = \frac{ ( c^2 \dot \theta^2 + \dot c^2 ) l m_0 m_1 + (l m_0^2 + 2 l m_0 m_1 + l m_1^2) (\dot x_{cm}^2 + \dot y_{cm}^2) + F ( l^2 - c^2 )(m_0 + m_1) }{2 l (m_0 + m_1)} }$ $\displaystyle{ = \frac{ ( c^2 \dot \theta^2 + \dot c^2 ) l m_0 m_1 + l (m_0 + m_1)^2 (\dot x_{cm}^2 + \dot y_{cm}^2) + F ( l^2 - c^2 )(m_0 + m_1) }{2 l (m_0 + m_1)} }$ $\displaystyle{ = \frac{1}{2} ( c^2 \dot \theta^2 + \dot c^2 ) \mu + \frac{1}{2} (m_0 + m_1) (\dot x_{cm}^2 + \dot y_{cm}^2) + \frac{1}{2l} F ( l^2 - c^2 ) }$


(show-expression
(((Lagrange-equations
(L-cm 'm_0 'm_1 'l))
(up (literal-function 'x_cm)
(literal-function 'y_cm)
(literal-function 'theta)
(literal-function 'c)
(literal-function 'F)))
't)) (show-expression
(((Lagrange-equations
(L-cm 'm_0 'm_1 'l))
(up (literal-function 'x_cm)
(literal-function 'y_cm)
(literal-function 'theta)
(lambda (t) 'l)
(literal-function 'F)))
't)) [guess]

— Me@2021-09-17 06:35:51 AM

.

.

# Meta numbers

Numbers are meta objects.

Infinities are meta numbers.

— Me@2017-02-03 05:20:51 PM

.

.

These are what I kept reminding myself during my Age of University.

1. I think, therefore I am

A lot of times, something seems to be a deadend.

However, once I have faith that a solution may be possible and start to think, most of the time, that “deadline” is actually resolvable.

2. Nothing Less

A lot of times, an action seems to be insignificant.

However, actually, every action may have infinite consequences, especially when you cultivate it.

A key to get a meaningful life is to dare to ask other people or yourself for answers, advice, or favours, when necessary.

Don’t be afraid to ask, as long as a request is ethical, reasonable, and polite.

— Me@2011.08.17

— Me@2021-09-11

.

.

# 相聚零刻 2.1

.

.

— Me@2021-07-26 05:12:55 PM

— Me@2021-09-07 05:15:56 PM

.

.

# 2.10 A spacetime orbifold in two dimensions, 3

A First Course in String Theory

.

(b) Draw a spacetime diagram, indicate the $\displaystyle{x^+}$ and $\displaystyle{x^-}$ axes, and sketch the family of curves $\displaystyle{x^+ x^- = a^2}$,

where $\displaystyle{a > 0}$ is a real constant that labels the various curves.

~~~ — Me@2021-08-31 08:42:40 PM

.

.

# Visualizing higher dimensions, 2

Geometry is global.

Space is what we can see at once.

Dynamics is local.

Time is what we cannot see at once.

— Me@2017-02-07 10:11:34 PM

.

If we could see, for example, several minutes at once, that several minutes would become a spatial dimension.

In other words, that dimension is visualized for us.

— Me@2017-02-03 07:31:25 AM

.

.

# Broken

A: Before meeting you, my life is chaotic.

M: It is because you missed, me. With me, you are completed.

— Me@2021-06-07 06:16:29 PM

— Me@2021-08-22 05:40:53 PM

.

.

# 上山尋寶, 2

.

— Me@2021-08-16 04:31:19 PM

.

.

# 1986, 2

— Me@2021-08-13 04:52:08 PM

.

.

# Ex 1.21 The dumbbell, 3.2

Structure and Interpretation of Classical Mechanics

.

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

~~~

[guess] \displaystyle{ \begin{aligned} m_0 \ddot y_0 &= F \sin \theta \\ m_0 \ddot x_0 &= F \cos \theta \\ m_1 \ddot y_1 &= -F \sin \theta \\ m_1 \ddot x_1 &= -F \cos \theta \\ \end{aligned}} \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} \ddot y_{cm} &= \frac{F \sin \theta - F \sin \theta}{m_0 + m_1} = 0 \\ \ddot x_{cm} &= \frac{F \cos \theta - F \cos \theta}{m_0 + m_1} = 0 \\ \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 \\ 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}} \displaystyle{ \begin{aligned} x_1 - x_0 &= \frac{m_0}{M} c(t) \cos \theta + \frac{m_1}{M} c(t) \cos \theta \\ &= c(t) \cos \theta \\ \end{aligned}} \displaystyle{ \begin{aligned} \dot x_1 - \dot x_0 &= \dot c(t) \cos \theta - c(t) \dot \theta \sin \theta \\ \end{aligned}} \displaystyle{ \begin{aligned} \ddot x_1 - \ddot x_0 &= \ddot c(t) \cos \theta - \dot c(t) \dot \theta \sin \theta - \dot c(t) \dot \theta \sin \theta - c(t) \ddot \theta \sin \theta - c(t) \dot \theta^2 \cos \theta\\ \end{aligned}} \displaystyle{ \begin{aligned} y_1 - y_0 &= c(t) \sin \theta \\ \dot y_1 - \dot y_0 &= \dot c(t) \sin \theta + c(t) \dot \theta \cos \theta \\ \ddot y_1 - \ddot y_0 &= \ddot c(t) \sin \theta + \dot c(t) \dot \theta \cos \theta + \dot c(t) \dot \theta \cos \theta + c(t) \ddot \theta \cos \theta - c(t) \dot \theta^2 \sin \theta \\ \end{aligned}}

. \displaystyle{ \begin{aligned} m_0 \ddot y_0 &= F \sin \theta \\ m_0 \ddot x_0 &= F \cos \theta \\ m_1 \ddot y_1 &= -F \sin \theta \\ m_1 \ddot x_1 &= -F \cos \theta \\ \end{aligned}}

When $\displaystyle{\dot c(t) = 0}$ and $\displaystyle{\ddot c(t) = 0}$, \displaystyle{ \begin{aligned} \ddot x_1 - \ddot x_0 &= - c(t) \ddot \theta \sin \theta - c(t) \dot \theta^2 \cos \theta \\ - \left( \frac{1}{m_1} + \frac{1}{m_1} \right) F \cos \theta &= - c(t) \ddot \theta \sin \theta - c(t) \dot \theta^2 \cos \theta\\ \end{aligned}} \displaystyle{ \begin{aligned} \ddot y_1 - \ddot y_0 &= ... \\ - \left( \frac{1}{m_1} + \frac{1}{m_0} \right) F \sin \theta &= c(t) \ddot \theta \cos \theta - c(t) \dot \theta^2 \sin \theta \\ \end{aligned}}

. \displaystyle{ \begin{aligned} \tan \theta &= \frac{ c(t) \ddot \theta \cos \theta - c(t) \dot \theta^2 \sin \theta }{- c(t) \ddot \theta \sin \theta - c(t) \dot \theta^2 \cos \theta} \\ &= \frac{ \ddot \theta - \dot \theta^2 \tan \theta }{- \ddot \theta \tan \theta - \dot \theta^2} \\ \end{aligned}} \displaystyle{ \begin{aligned} \tan \theta \left( - \ddot \theta \tan \theta - \dot \theta^2 \right) &= \ddot \theta - \dot \theta^2 \tan \theta \\ &... \\ 0 &= \ddot \theta (1 + \tan^2 \theta) \\ \ddot \theta &= 0 \\ \end{aligned}}

. \displaystyle{ \begin{aligned} - \left( \frac{1}{m_1} + \frac{1}{m_0} \right) F \sin \theta &= c(t) \ddot \theta \cos \theta - c(t) \dot \theta^2 \sin \theta \\ - \left( \frac{1}{m_1} + \frac{1}{m_1} \right) F \cos \theta &= - c(t) \ddot \theta \sin \theta - c(t) \dot \theta^2 \cos \theta\\ \end{aligned}}

Let $\displaystyle{\frac{1}{\mu} = \left( \frac{1}{m_1} + \frac{1}{m_1} \right)}$ and since $\displaystyle{\ddot \theta = 0}$, \displaystyle{ \begin{aligned} - \frac{1}{\mu} F \sin \theta &= - c(t) \dot \theta^2 \sin \theta \\ - \frac{1}{\mu} F \cos \theta &= - c(t) \dot \theta^2 \cos \theta\\ \end{aligned}}

Since $\displaystyle{\sin \theta}$ and $\displaystyle{\cos \theta}$ cannot be both zero at the same time, \displaystyle{ \begin{aligned} - \frac{1}{\mu} F &= - c(t) \dot \theta^2 \\ \end{aligned}}

Put $\displaystyle{c(t) = l}$, \displaystyle{ \begin{aligned} \frac{1}{\mu} F &= l \dot \theta^2 \\ \dot \theta^2 &= \frac{1}{l \mu} F \\ \end{aligned}}

[guess]

— Me@2021-08-08 05:41:21 PM

.

.

# Logical arrow of time, 9.4

The second law of thermodynamics’ derivation (Ludwig Boltzmann’s H-theorem) is with respect to an observer.

How does an observer keep losing microscopic information about a system?

— Me@2017-02-12 07:37:54 PM

.

This drew the objection from Loschmidt that it should not be possible to deduce an irreversible process from time-symmetric dynamics and a time-symmetric formalism: something must be wrong (Loschmidt’s paradox).

The resolution (1895) of this paradox is that the velocities of two particles after a collision are no longer truly uncorrelated. By asserting that it was acceptable to ignore these correlations in the population at times after the initial time, Boltzmann had introduced an element of time asymmetry through the formalism of his calculation.

— Wikipedia on Molecular chaos

.

Physical entropy’s value is with respect to an observer.

— Me@2017-02-12 07:37:54 PM

.

This “paradox” can be explained by carefully considering the definition of entropy. In particular, as concisely explained by Edwin Thompson Jaynes, definitions of entropy are arbitrary.

As a central example in Jaynes’ paper points out, one can develop a theory that treats two gases as similar even if those gases may in reality be distinguished through sufficiently detailed measurement. As long as we do not perform these detailed measurements, the theory will have no internal inconsistencies. (In other words, it does not matter that we call gases A and B by the same name if we have not yet discovered that they are distinct.) If our theory calls gases A and B the same, then entropy does not change when we mix them. If our theory calls gases A and B different, then entropy does increase when they are mixed. This insight suggests that the ideas of “thermodynamic state” and of “entropy” are somewhat subjective.

— Wikipedia on The mixing paradox

.

.

Turn my life into an ideal graduate school. Be my own professor.

— Me@2011.05.26

.

Doing real research, doing real teaching.

— Me@2021-08-06 05:03:16 PM

.

.

# To realize is to realize, 1.3

.

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?

2. god or no god

.

Whether “god” exists or not depends on your definition of the word “god”.

.

In some definitions, god does not exist, because of the definitions’ self-contradictory nature. For example, god is good but he wants you to suffer for no reason.

.

In some definitions, god is possible to exist. For example, a god is any being that has a higher level of consciousness than a human being.

We can say that a human being’s consciousness is higher than a dog’s, in the sense that a human can understand things that a dog cannot. For example, most dogs do not understand what a computer is.

Similarly, a dog’s consciousness is higher than an ant’s. An ant’s consciousness is higher than a tree’s. A tree’s consciousness is higher than a rock’s.

In the opposite direction, it is highly possible that in this universe, there are beings that have a higher level of consciousness than human beings. It is highly unlikely that human beings are the highest beings.

Even within the human being species itself, different people can have different levels of consciousness. Even within a single person’s lifetime, one can be at different levels at different ages.

.

In some definitions, god can exist. You become that god.

In those cases, whether god exists or not depends on whether YOU are willing to become that god, taking up his responsibilities.

— Me@2021-07-26 05:49:53 PM

.

.

# 物理一個字：力學第一步

This is my first created video.

Please click, like, share, and subscribe!

— Me@2021-06-07 05:00:34 PM

.

.