11 done, source code in footer

LL1/11-motion-in-one-dimension.md

@@ -3,7 +3,6 @@ title: 11. Motion in one dimension
 ---
 
 The motion of a system having one degree of freedom is said to take place in one dimension. The most general form of the Lagrangian of such a system in fixed external conditions is
-L = 1a(q)i2-U(q),
 
 ```load
 LL1/11.1
@@ -22,41 +21,22 @@ The equations of motion corresponding to these Lagrangians can be integrated in
 LL1/11.3
 ```
 
-The two arbitrary constants in the solution of the equations of motion are
-here represented by the total energy E and the constant of integration.
-Since the kinetic energy is essentially positive, the total energy always
-exceeds the potential energy, i.e. the motion can take place only in those
-regions of space where U(x) < E. For example, let the function U(x) be
-of the form shown in Fig. 6 (p. 26). If we draw in the figure a horizontal
-line corresponding to a given value of the total energy, we immediately find
-the possible regions of motion. In the example of Fig. 6, the motion can
-occur only in the range AB or in the range to the right of C.
+The two arbitrary constants in the solution of the equations of motion are here represented by the total energy E and the constant of integration.
+Since the kinetic energy is essentially positive, the total energy always exceeds the potential energy, i.e. the motion can take place only in those regions of space where U(x) < E. For example, let the function U(x) be of the form shown in 'TODO Fig. 6 (p. 26)'. If we draw in the figure a horizontal line corresponding to a given value of the total energy, we immediately find the possible regions of motion. In the example of Fig. 6, the motion can occur only in the range AB or in the range to the right of C.
 The points at which the potential energy equals the total energy,
-U(x) = E,
-(11.4)
-give the limits of the motion. They are turning points, since the velocity there
-is zero. If the region of the motion is bounded by two such points, then the
-motion takes place in a finite region of space, and is said to be finite. If the
-region of the motion is limited on only one side, or on neither, then the
-motion is infinite and the particle goes to infinity.
 
-A finite motion in one dimension is oscillatory, the particle moving re-
-peatedly back and forth between two points (in Fig. 6, in the potential well
-AB between the points X1 and x2). The period T of the oscillations, i.e. the
-time during which the particle passes from X1 to X2 and back, is twice the time
-from X1 to X2 (because of the reversibility property, §5) or, by (11.3),
-T(E) =
-(11.5)
+```load
+LL1/11.4
+```
 
-where X1 and X2 are roots of equation (11.4) for the given value of E. This for-
-mula gives the period of the motion as a function of the total energy of the
-particle.
-U
-A
-B
-C
-U=E
-x,
-X2
-X
-FIG. 6
+give the limits of the motion. They are *turning points*, since the velocity there is zero. If the region of the motion is bounded by two such points, then the motion takes place in a finite region of space, and is said to be *finite*. If the region of the motion is limited on only one side, or on neither, then the
+motion is *infinite* and the particle goes to infinity.
+
+A finite motion in one dimension is oscillatory, the particle moving repeatedly back and forth between two points (in Fig. 6, in the potential well AB between the points X1 and x2). The period T of the oscillations, i.e. the time during which the particle passes from X1 to X2 and back, is twice the time
+from X1 to X2 (because of the reversibility property, `LL1/5`) or, by `LL1/11.3`),
+
+```load
+LL1/11.5
+```
+
+where $x_1$ and $x_2$ are roots of equation `LL1/11.4` for the given value of $E$. This formula gives the period of the motion as a function of the total energy of the particle.

LL1/7-momentum.md

@@ -45,7 +45,7 @@ The additivity of the momentum is evident. Moreover, unlike the energy, the mome
 
 The three components of the momentum vector are all conserved only in the absence of an external field. The individual components may be conserved even in the presence of a field, however, if the potential energy in the field does not depend on all the Cartesian co-ordinates. The mechanical properties of the system are evidently unchanged by a displacement along the axis of a co-ordinate which does not appear in the potential energy, and so the corresponding component of the momentum is conserved. For example, in a uniform field in the z-direction, the $x$ and $y$ components of momentum are conserved.
 
-The equation `LL1/7.1` has a simple physical meaning. The derivative $\partialL/\partial\v{r}_a = - \partial U\partial\v{r}_a$ is the force $\v{F}_a$ acting on the $a$th particle. Thus equation `LL1/7.1` signifies that the sum of the forces on all the particles in a closed system is zero:
+The equation `LL1/7.1` has a simple physical meaning. The derivative $\partial L/\partial\v{r}_a = - \partial U\partial\v{r}_a$ is the force $\v{F}_a$ acting on the $a$th particle. Thus equation `LL1/7.1` signifies that the sum of the forces on all the particles in a closed system is zero:
 
 ```load
 LL1/7.4

LL1/equations/11.1.tex

@@ -0,0 +1 @@
+L=\mfrac{1}{2}a(q)\dot{q}^2-U(q)

LL1/equations/11.2.tex

@@ -0,0 +1 @@
+L=\mfrac{1}{2}m\dot{x}^2-U(x)

LL1/equations/11.3.tex

@@ -0,0 +1 @@
+t=\sqrt{\mfrac{1}{2}m}\int\frac{\dd{x}}{\sqrt{E-U(x)}}+\text{constant}

LL1/equations/11.4.tex

@@ -0,0 +1 @@
+U(x)=E,

LL1/equations/11.5.tex

@@ -0,0 +1 @@
+T(E)=\sqrt{2m}\int_{x_1(e)}^{x_2(e)}\frac{\dd{x}}{\sqrt{E-U(x)}}

LL1/toc.md

@@ -17,13 +17,13 @@ II.  CONSERVATION LAWS
 9. [Angular momentum](9-angular-momentum.html)
 10. [Mechanical similarity](10-mechanical-similarity.html)
 
+III.  INTEGRATION OF THE EQUATIONS OF MOTION
+11. [Motion in one dimension](11-motion-in-one-dimension.html)
 
 <span style="background-color: yellow; color: white: width: 100%;">
 🚧 WORK IN PROGRESS BELOW THIS POINT 🚧
 </span>
 
-III.  INTEGRATION OF THE EQUATIONS OF MOTION
-11. [Motion in one dimension](11-motion-in-one-dimension.html)
 12. [Determination of the potential energy from the period of oscillation](12-determination-of-the-potential-energy-from-the-period-of-oscillation.html)
 13. [The reduced mass](13-the-reduced-mass.html)
 14. [Motion in a central field](14-motion-in-a-central-field.html)

Makefile

@@ -9,7 +9,8 @@ out/%.html: %.md
 	mkdir -p $(dir $@)
 	pandoc	\
 		-s --filter tools/pf-filter.py \
-		--template tools/paragraph-template.html \
+		--template tools/template.html \
+		--metadata goatcounter=${GOATCOUNTER} \
 		--mathjax=https://cdn.jsdelivr.net/npm/mathjax@3.1/es5/tex-mml-chtml.js \
 		-f markdown+pipe_tables \
 		--resource-path .:equations \
@@ -26,8 +27,6 @@ clean:
 re: clean all
 
 deploy:
-# make re with PROD=true
 	PROD=true $(MAKE) re
 	. <(pass export/RCLONE_CONFIG/cloudflare-god)
-# copy because sync would remove physics-notes
 	rclone -v sync out/ r2:llcotp/

README.md

@@ -1,4 +1,9 @@
-## Mistakes
+## Todo
+
+ - figures
+ - problems
+
+## Mistakes?
 
 LL2. expression before 9.10, denominator is ds not sqrt(ds)
 LL3. between 4.6 and 4.7, I think that fg-gf is anti-Hermitian iff f and g are Hermitian

assets/favicon.svg

@@ -1,15 +0,0 @@
-<?xml version="1.0" standalone="no"?>
-<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 20010904//EN"
- "http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd">
-<svg version="1.0" xmlns="http://www.w3.org/2000/svg"
- width="52.000000pt" height="75.000000pt" viewBox="0 0 52.000000 75.000000"
- preserveAspectRatio="xMidYMid meet">
-
-<g transform="translate(0.000000,75.000000) scale(0.100000,-0.100000)"
-fill="#000000" stroke="none">
-<path d="M342 648 c-24 -29 -49 -130 -87 -343 -19 -107 -33 -155 -46 -155 -5
-0 -8 3 -7 7 2 5 0 9 -4 11 -5 1 -8 0 -8 -3 0 -3 0 -9 0 -15 0 -5 9 -10 20 -10
-32 0 48 47 90 262 40 203 59 271 67 236 6 -23 26 -23 21 -1 -4 22 -31 28 -46
-11z"/>
-</g>
-</svg>

tools/pf-filter.py

@@ -44,7 +44,7 @@ def action(elem, doc):
             slug = slug.replace(".md", ".html")
             html = f'''
             <a href="{slug}">
-            §{pnum} {title}
+            §{title}
             </a>
             '''
             elem = pf.RawInline(html, format='html')

tools/template.html

@@ -4,7 +4,7 @@
 <head>
   <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" type="text/javascript"></script>
   <link rel="stylesheet" type="text/css" href="/assets/style.css">
-  <link rel="icon" type="image/x-icon" href="/assets/favicon.svg">
+  <link rel="icon" type="image/x-icon" href="/assets/favicon.ico">
   <a href="toc.html">index</a>
   <meta charset="utf-8" />
   $if(goatcounter)$
@@ -64,7 +64,15 @@ $endif$
 </body>
 <hr>
 <footer>
-  <a href="https://github.com/jzck/llcotp">source code</a>
+  source code:
+  <ul>
+  $for(sourcefile)$
+  <li>
+    <a href="https://github.com/jzck/llcotp/blob/master/${sourcefile}">${sourcefile}</a>
+    (<a href="https://github.com/jzck/llcotp/edit/master/${sourcefile}">edit</a>)
+  </li>
+  $endfor$
+  </ul>
 </footer>
 </div>
 </html>