scmutils, 3

Posted on March 12, 2020 by rpflee

Structure and Interpretation of Classical Mechanics

Scheme Mechanics Installation for GNU/Linux

This post assumes that you have already installed the scmutils library and been able to open it using the standard editor Emacs.

If not, go to the bottom of this post to click the category scmutils, so that you can see all the posts in this scmutils series.

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Then go to the post “scmutils, 2.3.2” to follow the installation and setup instructions.

After installing and setting up the scmutils library, you can start to use it.

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However, what if you want to close the Emacs editor? How to save your scheme program before closing Emacs?

By default, you cannot. So I have written a small program to help. Here is the installation instruction:

1. Go to the end of the .emacs file. Add the following code, if it does not already exist:


(defun mechanics()
  (interactive)
  (run-scheme
   "/usr/local/scmutils/mit-scheme/bin/scheme --library
/usr/local/scmutils/mit-scheme/lib"
  ))

2. Add the following code:


(fset 'set-working-dir
      (lambda (&optional arg) "Keyboard macro."
	(interactive "p")
	(kmacro-exec-ring-item
	 (quote ("(set-working-directory-pathname! 
                  \"~/Documents/\")\n" 0 "%d")) arg)))
 
(fset 'load-scm
      (lambda (&optional arg) "Keyboard macro."
	(interactive "p")
	(kmacro-exec-ring-item
	 (quote ("(load \"tt.scm\")" 0 "%d")) arg)))
 
(defun mechan ()
  (interactive)
  (split-window-below)
  (windmove-down)
  (mechanics)
  (set-working-dir)
  (comint-send-input)
  (windmove-up)			
  (find-file "~/Documents/tt.scm")
  (end-of-buffer)
  (windmove-down)
  (cond ((file-exists-p "~/Documents/tt.scm")
	 (interactive)
	 (load-scm)
	 (comint-send-input)))
  (windmove-up)
)
 
(defun cxce ()
  (interactive)
  (save-buffer)
  (windmove-down)
  (load-scm)
  (comint-send-input)
  (windmove-up)
)
 
(global-set-key (kbd "C-x C-e") 'cxce)

3. Close Emacs. Re-open it.

4. Type the command

M-x mechan

The command M-x means pressing the Alt key and x together. Then type the word mechan.

5. You will see the Emacs editor is split into two windows, one up and one down.

The lower window is the scheme environment. You can type a line of code and the press Enter to execute it.

The upper window is the editor. You can type multiple lines of code and the type

C-x C-e

to execute it. The command C-x C-e means pressing Ctrl and x together and then Ctrl and e.

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6. Your scheme code is saved to the following file

"~/Documents/tt.scm"

In case you need to backup your code, backup this file.

— Me@2020-03-10 10:59:45 PM

scmutils, 2.3.2

Posted on February 22, 2020 by rpflee

Scheme Mechanics Installation for GNU/Linux

Steps:

1. The following steps are tested in Ubuntu 18.04. Prepare Ubuntu 18.04 if you can.

Note: Since the installation of the library scmutils requires the root access of your Linux system, please do NOT use it on your working computer. Instead, create an isolated virtual machine to use it.

2. Go to the bottom of this post to click the category scmutils, so that you can see all the posts in this scmutils series.

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3. Go to the post titled “scmutils, 2.3” to download

scmutils-20160827-x86-64-gnu-linux

4. Although the official installation guide advises you to install “MIT/GNU Scheme system” before installing scmutils, you do NOT need to install “MIT/GNU Scheme system” at all.

5. Unzip the file scmutils-20160827-x86-64-gnu-linux.

In the following, if you need to copy any commands or programming codes, remember that any number on the left of the vertical green line is NOT part of the code.

6. Run the command

tar xzf scmutils-20160827-x86-64-gnu-linux.tar.gz

to further extract the file.

-x — extract files from an archive;
-f — specify the archive’s name;
-v — show a list of processed files.

— Wikipedia on tar (computing)

Then two folders will be created: bin and scmutils.

7. Run the command

cd bin

to go into the folder.

You will see a file called mechanics.

8.1 Run the command

mechanics

You will get the error

mechanics: command not found

8.2 Instead, you should run the command

./mechanics

to specify that the file mechanics is actually in the current folder.

You will get the error

./mechanics: line 16: exec: xterm: not found

It is because your Linux system has not the program xterm yet.

8.3 Run the following command to install it.

sudo apt-get install xterm

8.4 Run the command again:

./mechanics

There will be an xterm window popup, but with an error message inside:

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That means you should move the two folders, bin and scmutils, to the pre-defined locations.

9.1 Run the command to move the folder scmutils to its pre-defined location:

mv scmutils /usr/local/

You will get the error

mv: cannot move 'scmutils' to '/usr/local/scmutils': 
Permission denied

9.2 Try again by

sudo mv scmutils /usr/local/

10.1 Go inside the folder bin.

10.2 Move its content to the pre-defined location by this command:

sudo mv mechanics /usr/local/bin/

11. Run the command

mechanics

Then you will see the Edwin window is opened. That means, in theory, you system has successfully installed the scmutils library. You can use it within the Edwin window if you like.

However, in practice, it is difficult, because it provides no syntax-highlighting. Also, you cannot use mouse in the Edwin window, so if you want to copy and paste a command or a series of commands, there will be no obvious way to achieve that.

So I suggest you to use the standard Emacs as the editor instead.

12.1 If you do not know Emacs, learn its basics.

12.2 Also, learn how to open Emacs’ initialization file, which has the filename

.emacs

After opening the file, you will see that it is just a text file.

12.3 Go to the end of the .emacs file. Add the following code:

(defun mechanics()
  (interactive)
  (run-scheme
     "/usr/local/scmutils/mit-scheme/bin/scheme --library
     /usr/local/scmutils/mit-scheme/lib"))

12.4 Save the file. Close Emacs. Then re-open Emacs.

12.5 Within Emacs, type the command

M-x mechanics

M-x means that while the Alt key is pressed down, press also x. Then type the word mechanics.

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12.6 Type the command

(+ 1 1)

to test the system.

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13. If you want to access your last command without re-typing it, type the command

M-p

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— Me@2020-02-22 06:25:47 PM

scmutils, 2.3

Posted on February 10, 2020 by rpflee

Within the MIT Scheme environment, it is not the original command line (bash) anymore. I can neither repeat the last command by just pressing the up key once, nor select the last command by mouse in order to copy it.

So I think I have to use an older version of scmutils.

However, this method is not easy to implement, because the author’s website does not provide an older version. Luckily, I have found an old version in my computer. You can download it here:

scmutils-20160827-x86-64-gnu-linux

— Me@2020-02-09 10:27:32 PM

scmutils, 2.2

Posted on January 27, 2020 by rpflee

Either use an older version of scmutils in order to follow the previous instructions for setting up Emacs for scmutils, or give up using Emacs for scmutils for the time being.

Using command line is the best way to go, so far.

The “using command line” method does not really work.

d_2020_01_27__16_12_48_PM_

Within the MIT Scheme environment, it is not the original command line (bash) anymore. I can neither repeat the last command by just pressing the up key once, nor select the last command by mouse in order to copy it.

So I think I have to use an older version of scmutils.

— Me@2020-01-26 07:38:31 PM

scmutils, 2

Posted on December 29, 2019 by rpflee

The original method of setting up Emacs for scmutils does not work anymore if you uses the newest (August 2019) version of scmutils, because its installation directories are not the same as those in the previous version.

Either use an older version of scmutils in order to follow the previous instructions for setting up Emacs for scmutils, or give up using Emacs for scmutils for the time being.

Using command line is the best way to go, so far.

Do not use the Edwin editor, since you cannot easily run, edit, or copy existing lines of code, unless you are familiar with it. I do not like it anyway, because after all, it does not provide syntax highlighting.

— Me@2019-12-28 07:50:32 PM

Varying the action, 2.2

Posted on December 15, 2019 by rpflee

…

$\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} (D L \circ \Gamma[q]) \delta_\eta \Gamma[q] \\ \end{aligned}}$

…

$\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} [\partial_0 L (t, q, v), \partial_1 L (t, q, v), \partial_2 L (t, q, v)] (0, \eta(t), D\eta(t)) \\ \end{aligned}}$

There are two kinds of tuples: up tuples and down tuples. We write tuples as ordered lists of their components; a tuple is delimited by parentheses if it is an up tuple and by square brackets if it is a down tuple.

— Structure and Interpretation of Classical Mechanics

So $\displaystyle{\left[\partial_0 L (t, q, v), \partial_1 L (t, q, v), \partial_2 L (t, q, v)\right] (0, \eta(t), D\eta(t))}$ is really a dot product:

$\displaystyle{ \begin{aligned} & \int_{t_1}^{t_2} (D L \circ \Gamma[q]) \delta_\eta \Gamma[q] \\ &= \int_{t_1}^{t_2} [\partial_0 L (t, q, v), \partial_1 L (t, q, v), \partial_2 L (t, q, v)] (0, \eta(t), D\eta(t)) \\ &= \int_{t_1}^{t_2} [\partial_1 L (t, q, v) \eta(t) + \partial_2 L (t, q, v) D\eta(t)] \\ \end{aligned}}$

— Me@2019-12-14 06:11:22 PM

Varying the action, 2.1

Posted on October 13, 2019 by rpflee

Equation (1.28):

$\displaystyle{S[q](t_1, t_2) = \int_{t_1}^{t_2} L \circ \Gamma[q]}$

Equation (1.30):

$\displaystyle{h[q] = L \circ \Gamma[q]}$

$\displaystyle{\delta_\eta S[q](t_1, t_2) = \int_{t_1}^{t_2} \delta_\eta h[q]}$

Let $\displaystyle{F}$ be a path-independent function and $\displaystyle{g}$ be a path-dependent function; then

$\displaystyle{\delta_\eta h[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]~~~~~\text{with}~~~~~h[q] = F \circ g[q].~~~~~(1.26)}$

$\displaystyle{\delta_\eta F \circ g[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]}$

— 1.5.1 Varying a path

— Structure and Interpretation of Classical Mechanics

$\displaystyle{ \begin{aligned} &\delta_\eta S[q] (t_1, t_2) \\ &= \int_{t_1}^{t_2} \delta_\eta \left( L \circ \Gamma[q] \right) \\ \end{aligned}}$

Assume that $\displaystyle{L}$ is a path-independent function, so that we can use Eq. 1.26:

$\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} (D L \circ \Gamma[q]) \delta_\eta \Gamma[q] \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} (D L \circ \Gamma[q]) (0, \eta(t), D\eta(t)) \\ &= \int_{t_1}^{t_2} (D L \left[ \Gamma[q] \right]) (0, \eta(t), D\eta(t)) \\ \end{aligned}}$

Assume that $\displaystyle{L}$ is a path-independent function, so that any value of $\displaystyle{L}$ depends on the value of $\displaystyle{\Gamma}$ at that moment only, instead of depending on the whole path $\displaystyle{\Gamma}$ :

$\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} (D L (\Gamma[q])) (0, \eta(t), D\eta(t)) \\ &= \int_{t_1}^{t_2} (D L (t, q, v)) (0, \eta(t), D\eta(t)) \\ &= \int_{t_1}^{t_2} [\partial_0 L (t, q, v), \partial_1 L (t, q, v), \partial_2 L (t, q, v)] (0, \eta(t), D\eta(t)) \\ \end{aligned}}$

What kind of product is it here? Is it just a dot product? Probably not.

$\displaystyle{ \begin{aligned} &= \int_{t_1}^{t_2} [\partial_1 L (t, q, v) \eta(t) + \partial_2 L (t, q, v) D\eta(t)] \\ \end{aligned}}$

— Me@2019-10-12 03:42:01 PM

Introduction to Differential Equations

Posted on October 5, 2019 by rpflee

llamaz 1 hour ago [-]

I think the calculus of variations might be a better approach to introducing ODEs in first year.

You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of $\displaystyle{\frac{dy}{dx} = 0}$ , you end up with an ODE (viz. the Euler-Lagrange equation).

This also motivates Lagrange multipliers which are usually taught around the same time as ODEs. They are similar to the Hamiltonian, which is a synonym for energy and is derived from the Euler-Lagrange equations of a system.

Of course you would brush over most of this mechanics stuff in a single lecture (60 min). But now you’ve motivated ODEs and given the students a reason to solve ODEs with constant coefficients.

— Hacker News

2019.10.02 Wednesday ACHK

A Tale of Two L’s

Posted on September 4, 2019 by rpflee

Lagrange’s equations are traditionally written in the form

$\displaystyle{\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}} \right ) = \frac {\partial L}{\partial q}}$

or, if we write a separate equation for each component of $\displaystyle{q}$ , as

$\displaystyle{\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q^i}} \right ) = \frac {\partial L}{\partial q^i}}$

In this way of writing Lagrange’s equations the notation does not distinguish between $\displaystyle{L}$ , which is a real-valued function of three variables $\displaystyle{(t, q, \dot q)}$ , and $\displaystyle{L \circ \Gamma[q]}$ , which is a real-valued function of one real variable $\displaystyle{t}$ .

— Structure and Interpretation of Classical Mechanics

2019.09.04 Wednesday ACHK

Literal numbers

Posted on August 17, 2019 by rpflee

All primitive mathematical procedures are extended to be generic over
symbolic arguments. When given symbolic arguments, these procedures
construct a symbolic representation of the required answer. There are
primitive literal numbers. We can make a literal number that is
represented as an expression by the symbol “a” as follows:

(literal-number 'a)        ==>  (*number* (expression a))

The literal number is an object that has the type of a number, but its
representation as an expression is the symbol “a”.

(type (literal-number 'a))          ==>  *number*

(expression (literal-number 'a))    ==>  a

— SCMUTILS Reference Manual

2019.08.17 Saturday ACHK

Ex 1.7 Properties of $\delta$

Posted on June 25, 2019 by rpflee

Let $\displaystyle{F}$ be a path-independent function and $\displaystyle{g}$ be a path-dependent function; then

$\displaystyle{\delta_\eta h[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]~~~~~\text{with}~~~~~h[q] = F \circ g[q].~~~~~(1.26)}$

— 1.5.1 Varying a path

— Structure and Interpretation of Classical Mechanics

Prove that

$\displaystyle{\delta_\eta F \circ g[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]}$

~~~

$\displaystyle{RHS = \lim_{\Delta t \to 0} \left( \frac{F \circ g[q](t+\Delta t) - F \circ g[q](t)}{\Delta t} \right) \lim_{\epsilon \to 0} \left( \frac{g[q+\epsilon \eta]-g[q]}{\epsilon} \right)}$

$\displaystyle{ \begin{aligned} LHS &= \delta_\eta F \circ g[q] \\ &= \lim_{\epsilon \to 0} \left( \frac{F \circ g[q+\epsilon \eta]-F \circ g[q]}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F \left[ g[q+\epsilon \eta] \right] - F \left[ g[q] \right]}{\epsilon} \right) \\ \end{aligned}}$

Since $\displaystyle{F}$ is path-independent,

$\displaystyle{ \begin{aligned} LHS &= \lim_{\epsilon \to 0} \left( \frac{F \left( g[q+\epsilon \eta ] \right) - F \left( g[q] \right)}{\epsilon} \right) \\ \end{aligned}}$

Let $\displaystyle{ g[q+\epsilon \eta] = g + \Delta g}$ .

$\displaystyle{ \begin{aligned} LHS &= \lim_{\epsilon \to 0} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\ \end{aligned}}$

When $\displaystyle{ \epsilon \to 0}$ , $\displaystyle{ \Delta g \to 0 }$ .

$\displaystyle{ \begin{aligned} LHS &= \lim_{\substack{\epsilon \to 0 \\ \Delta g \to 0}} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\ &= \lim_{\Delta g \to 0} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]} \lim_{\epsilon \to 0} \frac{g[q + \epsilon \eta] - g[q]}{\epsilon} \right) \\ &= DF \left( g[q] \right) \delta_\eta g[q] \\ &= RHS \\ \end{aligned}}$

— Me@2019-06-24 10:55:28 PM

The Principle of Stationary Action

Posted on May 13, 2019 by rpflee

Varying the action

$\displaystyle{S[q](t_1, t_2) = \int_{t_1}^{t_2} L \circ \Gamma[q]}$

The Principle of Stationary Action

$\displaystyle{\delta_\eta S[q] (t_1, t_2) = 0}$

$\delta_\eta S[q] (t_1, t_2) = \int_{t_1}^{t_2} \delta_\eta L \circ \Gamma[q]$

— Structure and Interpretation of Classical Mechanics

2019.05.13 Monday ACHK

Ex 1.8 Implementation of $\delta$

Posted on May 5, 2019 by rpflee

$\displaystyle{ \begin{aligned} \delta_\eta f[q] &= \lim_{\epsilon \to 0} \left( \frac{f[q+\epsilon \eta]-f[q]}{\epsilon} \right) \\ \end{aligned}}$

The variation may be represented in terms of a derivative.

— Structure and Interpretation of Classical Mechanics

$\displaystyle{ \begin{aligned} g( \epsilon ) &= f[q + \epsilon \eta] \\ \delta_\eta f[q] &= \lim_{\epsilon \to 0} \left( \frac{g(\epsilon) - g(0)}{\epsilon} \right) \\ &= D g(0) \\ \end{aligned}}$

A lambda expression evaluates to a procedure. The environment in effect when the lambda expression is evaluated is remembered as part of the procedure; it is called the closing environment.

— Structure and Interpretation of Classical Mechanics

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

— Me@2019-05-05 10:47:46 PM

Varying a path

Posted on April 28, 2019 by rpflee

Suppose that we have a function $\displaystyle{f[q]}$ that depends on a path $\displaystyle{q}$ . How does the function vary as the path is varied? Let $\displaystyle{q}$ be a coordinate path and $\displaystyle{q + \epsilon \eta}$ be a varied path, where the function $\displaystyle{\eta}$ is a path-like function that can be added to the path $\displaystyle{q}$ , and the factor $\displaystyle{\epsilon}$ is a scale factor. We define the variation $\displaystyle{ \delta_\eta f[q]}$ of the function $\displaystyle{f}$ on the path $\displaystyle{q}$ by

$\displaystyle{\delta_\eta f [q] = \lim_{\epsilon \to 0} \left( \frac{f[q + \epsilon \eta] - f[q]}{\epsilon} \right)}$

The variation of $\displaystyle{f}$ is a linear approximation to the change in the function $\displaystyle{f}$ for small variations in the path. The variation of $\displaystyle{f}$ depends on $\displaystyle{\eta}$ .

— 1.5.1 Varying a path

— Structure and Interpretation of Classical Mechanics

Exercise 1.7. Properties of $\displaystyle{\delta}$

The meaning of $\displaystyle{\delta_\eta (fg)[q]}$ is

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

— Me@2019-04-27 07:02:38 PM

2019.04.27 Saturday ACHK

scmutils

Posted on April 7, 2019 by rpflee

In order to run the SICM code, you need to install the scmutils library. Just go to the official page to download the library and follow the official instructions to install it in a Linux operating system.

When you try to run it, your system may give the following error message:

/usr/local/bin/mechanics: line 16: exec: xterm: not found

If so, you should install the program xterm first.

Also, in case you like to use Emacs as editor, you can:

Just include the following in your .emacs file:

(defun mechanics ()
  (interactive)
  (run-scheme
    "ROOT/mit-scheme/bin/scheme --library ROOT/mit-scheme/lib"
  ))

Replace ROOT with the directory in which you installed the scmutils software. (Remember to replace it in both places. If it is installed differently on your system, just make sure the string has the form “/path/to/mit-scheme --library /path/to/scmutils-library“.) Restart emacs (or use C-x C-e to evaluate the mechanics defun), and launch the environment with the command M-x mechanics.

— Using GNU Emacs With SCMUtils

— Aaron Maxwell

In my Ubuntu 18.04, the paths are:

(defun mechanics()
  (interactive)
  (run-scheme
   "/usr/local/scmutils/mit-scheme/bin/scheme --library 
/usr/local/scmutils/mit-scheme/lib"
  ))

— Me@2019-04-07 02:52:50 PM

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

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

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

Path-distinguishing function, 2

Posted on February 24, 2019 by rpflee

$\displaystyle{\gamma(t)}$ = configuration path function

$\displaystyle{\mathcal{F} [\gamma]}$ = a function of time that measures some local property of the path

……….It may depend upon the value of the function $\displaystyle{\gamma}$ at that time

……….and the value of any derivatives of $\displaystyle{\gamma}$ at that time.

We can decompose $\mathcal{F} [\gamma]$ into two parts:

1. a part that measures some property of a local description

and

2. a part that extracts a local description of the path from the path function.

— 1.3 The Principle of Stationary Action

— Structure and Interpretation of Classical Mechanics

1. The function that measures the local property of the system depends on the particular physical system;

2. the method of construction of a local description of a path from a path is the same for any system.

$\displaystyle{ \begin{aligned} \mathcal{F} [\gamma] &= \mathcal{L} \circ \mathcal{T}[\gamma] \\ \mathcal{T} [\gamma] &= (t, \gamma (t), \mathcal{D} \gamma (t), ...) \end{aligned}}$

— 1.3 The Principle of Stationary Action

— Structure and Interpretation of Classical Mechanics

— Me@2019-02-22 11:46:50 PM

2019.02.24 Sunday ACHK

Path-distinguishing function

Posted on February 17, 2019 by rpflee

So we will try to arrange that the path-distinguishing function, constructed as an integral of a local property along the path, assumes a stationary value for any realizable path. Such a path-distinguishing function is traditionally called an action for the system. We use the word “action” to be consistent with common usage. Perhaps it would be clearer to continue to call it “path-distinguishing function,” but then it would be more difficult for others to know what we were talking about.

— 1.3 The Principle of Stationary Action

— Structure and Interpretation of Classical Mechanics

2019.02.17 Sunday ACHK

Physics Town

continues

Category Archives: SICM

scmutils, 3

scmutils, 2.3.2

scmutils, 2.3

scmutils, 2.2

scmutils, 2

Varying the action, 2.2

Varying the action, 2.1

Introduction to Differential Equations

A Tale of Two L’s

Literal numbers

Ex 1.7 Properties of $\delta$

The Principle of Stationary Action

Ex 1.8 Implementation of $\delta$

Varying a path

scmutils

Finding trajectories that minimize the action

Computing Actions

Lagrangians in generalized coordinates

Computing Actions

Exercise 1.4. Lagrangian actions

Generalized Coordinates

Configuration Spaces

Generalized Coordinates

Path-distinguishing function, 2

Path-distinguishing function