Ex 1.8.2.5 Implementation of $delta$

Structure and Interpretation of Classical Mechanics

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Verify that the operators $\displaystyle{D}$ (differentiation) and $\displaystyle{\delta}$ (variation) commute (Equation 1.27) using the scmutils software library:

$\displaystyle{D \delta_{\eta} f [q] = \delta_\eta g[q]}$ with $\displaystyle{g [q] = D ( f[q] )}$

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(define (((delta eta) f) q)
(define (g epsilon)
(f (+ q (* epsilon eta))))
((D g) 0))

(define q (literal-function 'q (-> Real (UP Real))))

(define eta (literal-function 'eta (-> Real (UP Real))))


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(define (f q)
(compose (literal-function 'f
(-> (UP Real (UP* Real) (UP* Real)) Real))
(Gamma q)))

(define (g q)
(compose (literal-function 'g
(-> (UP Real (UP* Real) (UP* Real)) Real))
(Gamma q)))


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(define (g q) (D (f q)))


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(define LHS ( (D (((delta eta) f) q)) 't))

(define RHS ((((delta eta) g) q) 't))


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(print-expression LHS)

(show-expression LHS)


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$\displaystyle{\partial_1 \partial_1 f \left( \begin{bmatrix} t \\ q(t) \\ Dq(t) \end{bmatrix} \right) D q(t) \eta(t) + D^2 q \partial_2 \partial_2 f D \eta + D^2 q \partial_1 \partial_2 f \eta + ... }$

$\displaystyle{... \partial_1 \partial_2 f D \eta D q + D^2 \eta \partial_2 f + \partial_0 \partial_2 f D \eta + \partial_0 \partial_1 f \eta + D \eta \partial_1 f}$

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(print-expression RHS)

(show-expression RHS)

(- LHS RHS)


— Me@2020-08-24 03:18:21 PM

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三一萬能俠, 2

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Theoretical Mechanics I

Theoretical Mechanics II

Differential Equations

Numerical Methods

Probability and Statistics

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How would it be possible, if salvation were ready to our hand, and could without great labour be found, that should be by almost all men neglected?

But all excellent things are as difficult as they are rare.

— Spinoza

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1. What is the happiest moment in your life ever?

2. Can you re-create that moment?

3. If not, why not?

— Me@2013.08.03

1. 到目前為止，哪個時刻是你最快樂的？

2. 你可以重造那個時刻嗎？

3. 如果不可以的話，為什麼不可以呢？

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（如果是最快樂的一刻，我選第一次見到我弟弟的那一刻。

— Me@2020-08-16 07:15:02 PM

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1990s, 3

— Me@2020-08-15 03:49:22 PM

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Problem 2.3b2

Prove that a metric tensor is symmetric.

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Assume $\displaystyle{\eta_{\alpha\beta} \neq \eta_{\beta\alpha}}$. Because it’s irrelevant what letter we use for our indices,

$\displaystyle{\eta_{\alpha\beta}dx^{\alpha}dx^{\beta} = \eta_{\beta\alpha}dx^{\beta}dx^{\alpha}}$.

Then

$\displaystyle{\eta_{\alpha\beta}dx^{\alpha}dx^{\beta} = \frac{1}{2}(\eta_{\alpha\beta}dx^{\alpha}dx^{\beta} + \eta_{\beta\alpha}dx^{\beta}dx^{\alpha}) = \frac{1}{2} (\eta_{\alpha\beta} + \eta_{\beta\alpha})dx^{\alpha}dx^{\beta}}$

So only the symmetric part of $\displaystyle{\eta_{\alpha\beta}}$ would survive the sum. As such we may as well take $\displaystyle{\eta_{\alpha\beta}}$ to be symmetric in its definition.

— edited Jun 15 ’15 at 22:48

— rob

— answered Jun 15 ’15 at 17:52

— FenderLesPaul

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— Why is the metric tensor symmetric?

— Physics StackExchange

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1.

$\displaystyle{\eta_{\alpha\beta}dx^{\alpha}dx^{\beta} = \eta_{\beta\alpha}dx^{\beta}dx^{\alpha}}$

means that

$\displaystyle{\sum_{\alpha, \beta} \eta_{\alpha\beta}dx^{\alpha}dx^{\beta}=\sum_{\alpha, \beta}\eta_{\beta\alpha}dx^{\beta}dx^{\alpha}}$

So in

$\displaystyle{\eta_{\alpha\beta}dx^{\alpha}dx^{\beta} = \eta_{\beta\alpha}dx^{\beta}dx^{\alpha}}$,

we cannot cancel out $\displaystyle{dx^{\alpha}dx^{\beta}}$ on both sides. In other words, we do NOT assume that $\displaystyle{\eta_{\alpha\beta} = \eta_{\beta\alpha}}$ in the first place.

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2.

$\displaystyle{\eta_{\alpha\beta}dx^{\alpha}dx^{\beta} = \frac{1}{2}(\eta_{\alpha\beta}dx^{\alpha}dx^{\beta} + \eta_{\beta\alpha}dx^{\beta}dx^{\alpha}) = \frac{1}{2} (\eta_{\alpha\beta} + \eta_{\beta\alpha})dx^{\alpha}dx^{\beta}}$

means that

$\displaystyle{\sum_{\alpha, \beta}\eta_{\alpha\beta}dx^{\alpha}dx^{\beta} = \frac{1}{2}\sum_{\alpha, \beta}(\eta_{\alpha\beta}dx^{\alpha}dx^{\beta} + \eta_{\beta\alpha}dx^{\beta}dx^{\alpha}) = \frac{1}{2} \sum_{\alpha, \beta}(\eta_{\alpha\beta} + \eta_{\beta\alpha})dx^{\alpha}dx^{\beta}}$

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3. “… only the symmetric part of $\displaystyle{\eta_{\alpha\beta}}$ would survive the sum” means that only the sum $\displaystyle{\left(\eta_{\alpha\beta} + \eta_{\beta\alpha}\right)}$ is physically meaningful.

— Me@2020-08-14 03:34:05 PM

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Spacetime rate

Every motion in space is also a motion in time.

The speed of light is the upper limit of the spacetime rate.

— Me@2012-04-28 12:42:08 PM

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An object cannot change this position without changing its time coordinate.

In short, there is no instantaneous motion.

— Me@2020-08-12 05:26:20 PM

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Nuclear power, 2

In Search of Stupidity, 2

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The exciting thing about market economies is that stupidity equals opportunity.

— Paul Graham

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politics ~ capitalize on stupidity [to get things done]

— Me@2011.06.24

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Stupidity is like nuclear power; it can be used for good or evil.

— Dilbert

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機遇創生論 1.6.3

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「通用知識」的意思是，每人日常也需要知道的東西，例如，健康、財政、人際、時間管理等。

As we get older, generic reading becomes less and less useful. We then gain new knowledge mostly by personal life experience and directed reading.

— paraphrasing John T. Reed

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— Me@2020-08-11 03:09:13 PM

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1990s, 2

— Me@2020-08-07 05:08:07 PM

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Ex 1.8.2.4 Implementation of $\delta$

Structure and Interpretation of Classical Mechanics

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Verify the product rule of variation (Equation 1.23) using the
scmutils software library:

$\displaystyle{\delta_\eta \left(f [q] g [q] \right) = \left( \delta_\eta f[q] \right) g[q] + f[q] \delta_\eta g[q]}$

~~~

(define (((delta eta) f) q)
(define (g epsilon)
(f (+ q (* epsilon eta))))
((D g) 0))

(define q (literal-function 'q (-> Real (UP Real))))

(define eta (literal-function 'eta (-> Real (UP Real))))


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(define (f q)
(compose (literal-function 'f
(-> (UP Real (UP* Real) (UP* Real)) Real))
(Gamma q)))

(define (g q)
(compose (literal-function 'g
(-> (UP Real (UP* Real) (UP* Real)) Real))
(Gamma q)))


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(define (f_times_g q) (* (f q) (g q)))


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(define LHS ((((delta eta) f_times_g) q) 't))

(define RHS (+ (* ((((delta eta) f) q) 't) ((g q) 't))
(* ((f q) 't) ((((delta eta) g) q) 't))))


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(print-expression LHS)

(show-expression LHS)


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$\displaystyle{g D \eta \partial_2 f + g \eta \partial_1 f + f D \eta \partial_2 g + f \eta \partial_1 g}$

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(print-expression RHS)

(show-expression RHS)

(- LHS RHS)


— Me@2020-08-06 07:23:27 PM

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The square root of the probability, 2

Mixed states, 4.2

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Superposition in quantum mechanics is a complex number superposition.

— Me@2017-08-02 02:56:23 PM

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Superposition in quantum mechanics is not a superposition of probabilities.

Instead, it is a superposition of probability amplitudes, which have complex number values.

Probability amplitude, in a sense, is the square root of probability.

— Me@2020-08-04 03:38:43 PM

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大腦物理性損毀不是比喻, 2

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— 李穎

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2020.08.04 Tuesday ACHK

太極滅世戰 2.1

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《輪椅中的宇宙》和《時間簡史》令我發現，我的人生目標是物理。

— Me@2020-07-27 07:33:45 AM

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