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
.emacsfile:
(defun mechanics ()
(interactive)
(run-scheme
"ROOT/mit-scheme/bin/scheme --library ROOT/mit-scheme/lib"
))
Replace
ROOTwith the directory in which you installed thescmutilssoftware. (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“.) Restartemacs(or useC-x C-eto evaluate the mechanics defun), and launch the environment with the commandM-xmechanics.— 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
.
.
2019.04.07 Sunday (c) All rights reserved by ACHK
takes a coordinate path; the function 
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}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cbegin%7Baligned%7D++%5Cmathcal%7BS%7D+%5B%5Cgamma%5D+%28t_1%2C+t_2%29+%26%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+%5Cmathcal%7BL%7D+%5Ccirc+%5Cmathcal%7BT%7D+%5B%5Cgamma%5D++%5C%5C+++S_%5Cchi+%5Bq%5D+%28t_1%2C+t_2%29+%26%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+L_%5Cchi+%5Ccirc+%5CGamma+%5Bq%5D++%5C%5C+++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\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}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cbegin%7Baligned%7D++%5Cmathcal%7BS%7D+%5B%5Cgamma%5D+%28t_1%2C+t_2%29++%26%3D+S_%5Cchi+%5B%5Cchi+%5Ccirc+%5Cgamma%5D+%28t_1%2C+t_2%29+%5C%5C++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{\begin{aligned} \mathcal{S} [\gamma] (t_1, t_2) &= S_\chi [\chi \circ \gamma] (t_1, t_2) \\ \end{aligned}}")
makes a procedure that represents a function of one argument that has no known properties other than the given symbolic name.
is the constant-velocity straight-line path of a free particle, such that 
and 
. Show that the action on the solution path is 
![\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}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cbegin%7Baligned%7D++S_%5Cchi+%5B%5Cgamma%5D+%28t_1%2C+t_2%29+%26%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+L_%5Cchi+%28t%2C+q%28t%29%2C+Dq%28t%29%29+dt+%5C%5C++%26%3D+%5Cint_%7Bt_2%7D%5E%7Bt_1%7D+%5Cfrac%7B1%7D%7B2%7D+m+v%5E2+dt+%5C%5C++%26%3D+%5Cfrac%7B1%7D%7B2%7D+m+v%5E2+%5Cint_%7Bt_2%7D%5E%7Bt_1%7D+dt+%5C%5C++%26%3D+%5Cfrac%7B1%7D%7B2%7D+m+v%5E2+%28t_2+-+t_1%29++%5C%5C++%26%3D+%5Cfrac%7B1%7D%7B2%7D+m+%28%5Cfrac%7Bx_2+-+x_1%7D%7Bt_2+-+t_1%7D%29%5E2+%28t_2+-+t_1%29++%5C%5C++%26%3D+%5Cfrac%7B1%7D%7B2%7D+m+%5Cfrac%7B%28x_2+-+x_1%29%5E2%7D%7Bt_2+-+t_1%7D+++%5C%5C+++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\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}}")
-dimensional configuration space can be parameterized by choosing a coordinate function 
that maps elements of the configuration space to 
-tuples of real numbers.
mapping time to configuration-space points.
mapping time to tuples of generalized coordinates.
takes the coordinate-free local tuple 
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.
 &= (t, q(t), Dq(t), ...) \\ \Gamma[q] &= \Xi_\chi \circ \mathcal{T}[\gamma] \\ \end{aligned} }](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D++%28t%2C+q%28t%29%2C+Dq%28t%29%2C+...%29+%26%3D+%5CXi_%5Cchi+%28t%2C+%5Cgamma%28t%29%2C+%5Cmathcal%7BD%7D+%5Cgamma%28t%29%2C+...%29+++++%5C%5C++%5C%5C++%5CGamma%5Bq%5D%28t%29+%26%3D+%28t%2C+q%28t%29%2C+Dq%28t%29%2C+...%29+%5C%5C++%5CGamma%5Bq%5D+%26%3D+%5CXi_%5Cchi+%5Ccirc+%5Cmathcal%7BT%7D%5B%5Cgamma%5D+%5C%5C+++%5Cend%7Baligned%7D+%7D&bg=ffffff&fg=333333&s=0 "\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} }")
= configuration path function![\displaystyle{\mathcal{F} [\gamma]}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cmathcal%7BF%7D+%5B%5Cgamma%5D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{\mathcal{F} [\gamma]}")
= a function of time that measures some local property of the path![\mathcal{F} [\gamma]](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BF%7D+%5B%5Cgamma%5D&bg=ffffff&fg=333333&s=0 "\mathcal{F} [\gamma]")
into two parts:![\displaystyle{ \begin{aligned} \mathcal{F} [\gamma] &= \mathcal{L} \circ \mathcal{T}[\gamma] \\ \mathcal{T} [\gamma] &= (t, \gamma (t), \mathcal{D} \gamma (t), ...) \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D++%5Cmathcal%7BF%7D+%5B%5Cgamma%5D+%26%3D+%5Cmathcal%7BL%7D+%5Ccirc+%5Cmathcal%7BT%7D%5B%5Cgamma%5D+%5C%5C++%5Cmathcal%7BT%7D+%5B%5Cgamma%5D+%26%3D+%28t%2C+%5Cgamma+%28t%29%2C+%5Cmathcal%7BD%7D+%5Cgamma+%28t%29%2C+...%29++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{ \begin{aligned} \mathcal{F} [\gamma] &= \mathcal{L} \circ \mathcal{T}[\gamma] \\ \mathcal{T} [\gamma] &= (t, \gamma (t), \mathcal{D} \gamma (t), ...) \end{aligned}}")
