deployed

LL1/1-generalized-co-ordinates.md

@@ -1,34 +1,18 @@
 ---
-title: Generalized co-ordinates
+title: 1. Generalized co-ordinates
 ---
 
-One of the fundamental concepts of mechanics is that of a particle[^1]. By this
+One of the fundamental concepts of mechanics is that of a particle[^1]. By this we mean a body whose dimensions may be neglected in describing its motion.  The possibility of so doing depends, of course, on the conditions of the problem concerned. For example, the planets may be regarded as particles in considering their motion about the Sun, but not in considering their rotation about their axes.
-we mean a body whose dimensions may be neglected in describing its motion.
-The possibility of so doing depends, of course, on the conditions of the prob-
-lem concerned. For example, the planets may be regarded as particles in
-considering their motion about the Sun, but not in considering their rotation
-about their axes.
 
-The position of a particle in space is defined by its radius vector $r$, whose
+The position of a particle in space is defined by its radius vector $r$, whose components are its Cartesian co-ordinates $x$, $y$, $z$. The derivative $\v{v} = \dd{\v{r}}/\dd{t}$. of $r$ with respect to the time $t$ is called the velocity of the particle, and the second derivative $\dd[2]{\v{r}}/\dd[2]{t}$ is its acceleration. In what follows we shall, as is customary, denote differentiation with respect to time by placing a dot above a letter: $\v{v} = \dot{\v{r}}$.
-components are its Cartesian co-ordinates $x$, $y$, $z$. The derivative $\v{v} = \dd{\v{r}}/\dd{t}$. of $r$ with respect to the time $t$ is called the velocity of the particle, and the second derivative $\dd[2]{\v{r}}/\dd[2]{t}$ is its acceleration. In what follows we shall, as is customary, denote differentiation with respect to time by placing a dot above a letter: $\v{v} = \dot{\v{r}}$.
 
-To define the position of a system of N particles in space, it is necessary to
+To define the position of a system of N particles in space, it is necessary to specify $N$ radius vectors, i.e. $3N$ co-ordinates. The number of independent quantities which must be specified in order to define uniquely the position of any system is called the number of degrees of freedom; here, this number is $3N$. These quantities need not be the Cartesian co-ordinates of the particles, and the conditions of the problem may render some other choice of co-ordinates more convenient. Any $S$ quantities $q_1$, $q_2$, ..., $q_s$ which completely define the position of a system with $S$ degrees of freedom are called generalised co-ordinates of the system, and the derivatives $q_i$ are called its generalised velocities.
-specify $N$ radius vectors, i.e. $3N$ co-ordinates. The number of independent
-quantities which must be specified in order to define uniquely the position of
-any system is called the number of degrees of freedom; here, this number is $3N$. These quantities need not be the Cartesian co-ordinates of the particles,
-and the conditions of the problem may render some other choice of co-
-ordinates more convenient. Any $S$ quantities $q_1$, $q_2$, ..., $q_s$ which completely define the position of a system with $S$ degrees of freedom are called generalised co-ordinates of the system, and the derivatives $q_i$ are called its generalised velocities.
 
 When the values of the generalised co-ordinates are specified, however, the "mechanical state" of the system at the instant considered is not yet determined in such a way that the position of the system at subsequent instants can be predicted. For given values of the co-ordinates, the system can have any velocities, and these affect the position of the system after an infinitesimal time interval $\dd{t}$.
 
-If all the co-ordinates and velocities are simultaneously specified, it is
+If all the co-ordinates and velocities are simultaneously specified, it is known from experience that the state of the system is completely determined and that its subsequent motion can, in principle, be calculated. Mathematically, this means that, if all the co-ordinates $q$ and velocities $\dot{q}$ are given at some instant, the accelerations $\ddot{q}$ at that instant are uniquely defined[^2].
-known from experience that the state of the system is completely determined
-and that its subsequent motion can, in principle, be calculated. Mathematic-
-ally, this means that, if all the co-ordinates $q$ and velocities $\dot{q}$ are given at some instant, the accelerations $\ddot{q}$ at that instant are uniquely defined[^2].
 
-The relations between the accelerations, velocities and co-ordinates are
+The relations between the accelerations, velocities and co-ordinates are called the equations of motion. They are second-order differential equations for the functions $q(t)$, and their integration makes possible, in principle, the determination of these functions and so of the path of the system.
-called the equations of motion. They are second-order differential equations
-for the functions $q(t)$, and their integration makes possible, in principle, the determination of these functions and so of the path of the system.
 
 [^1]: Sometimes called in Russian a material point
 [^2]: For brevity, we shall often conventionally denote by q the set of all the co-ordinates $q_1,q_2,...,q_s$ and similarly by $\dot{q}$ the set of all velocities.

LL1/10-mechanical-similarity.md

@@ -1,5 +1,5 @@
 ---
-title: Mechanical similarity
+title: 10. Mechanical similarity
 ---
 
 Multiplication of the Lagrangian by any constant clearly does not affect the equations of motion. This fact (already mentioned in `LL1/2`) makes possible, in a number of important cases, some useful inferences concerning the properties of the motion, without the necessity of actually integrating the equations.
@@ -12,7 +12,7 @@ LL1/10.1
 
 where $\alpha$ is any constant and $k$ the degree of homogeneity of the function.
 
-Let us carry out a transformation in which the co-ordinates are changed by a factor $\alpha$ and the time by a factor $\beta: \v{r}_a\rightarrow \alpha\v{r}_a, t\rightarrow \beta t$. Then all the velocities $\v{v}a = \dd{\v{r}}_a/\dd{t} are changed by a factor $\alpha/\beta$, and the kinetic energy by a factor
+Let us carry out a transformation in which the co-ordinates are changed by a factor $\alpha$ and the time by a factor $\beta: \v{r}_a\rightarrow \alpha\v{r}_a, t\rightarrow \beta t$. Then all the velocities $\v{v}_a = \dd{\v{r}}_a/\dd{t}$ are changed by a factor $\alpha/\beta$, and the kinetic energy by a factor
 $\alpha^2/\beta^2$. The potential energy is multiplied by $\alpha^k$. If $\alpha$ and $\beta$ are such that $\alpha^2/\beta^2 = \alpha^k$, i.e. $\beta = \alpha^{1-k/2}$, then the result of the transformation is to multiply the Lagrangian by the constant factor $\alpha^k$, i.e. to leave the equations of motion unaltered.
 
 A change of all the co-ordinates of the particles by the same factor corresponds to the replacement of the paths of the particles by other paths, geometrically similar but differing in size. Thus we conclude that, if the potential energy of the system is a homogeneous function of degree k in the (Cartesian) co-ordinates, the equations of motion permit a series of geometrically similar paths, and the times of the motion between corresponding points are in the ratio

LL1/10_problems.md

@@ -1,3 +1,6 @@
+---
+---
+
 PROBLEMS
 PROBLEM 1. Find the ratio of the times in the same path for particles having different
 masses but the same potential energy.

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

@@ -1,5 +1,5 @@
 ---
-title: Motion in one dimension
+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
@@ -13,7 +13,7 @@ where $a(q)$ is some function of the generalised co-ordinate $q$. In particular,
 if $q$ is a Cartesian co-ordinate ($x$, say) then
 
 ```load
-
+LL1/11.2
 ```
 
 The equations of motion corresponding to these Lagrangians can be integrated in a general form. It is not even necessary to write down the equation of motion; we can start from the first integral of this equation, which gives the law of conservation of energy. For the Lagrangian `LL1/11.2` (e.g.) we have $\mfrac{1}{2}m\dot{x}^2+U(x)=E$. This is a first-order differential equation, and can be inte- grated immediately. Since $\dd{x}/\dd{t} = \sqrt{2[E - U(x)]/m}$, it follows that

LL1/12-determination-of-the-potential-energy-from-the-period-of-oscillation.md

@@ -1,5 +1,5 @@
 ---
-title: Determination of the potential energy from the period of oscillation
+title: 12. Determination of the potential energy from the period of oscillation
 ---
 
 Let us consider to what extent the form of the potential energy U(x) of a

LL1/13-the-reduced-mass.md

@@ -1,5 +1,5 @@
 ---
-title: The reduced mass
+title: 13. The reduced mass
 ---
 
 A complete general solution can be obtained for an extremely important

LL1/14-motion-in-a-central-field.md

@@ -1,5 +1,5 @@
 ---
-title: Motion in a central field
+title: 14. Motion in a central field
 ---
 
 On reducing the two-body problem to one of the motion of a single body,

LL1/2-the-principle-of-least-action.md

@@ -1,5 +1,5 @@
 ---
-title: The principle of least action
+title: 2. The principle of least action
 ---
 
 The most general formulation of the law governing the motion of mech-

LL1/3-galileos-relativity-principle.md

@@ -1,5 +1,5 @@
 ---
-title: Galileo's relativity principle
+title: 3. Galileo's relativity principle
 ---
 
 In order to consider mechanical phenomena it is necessary to choose a frame of reference. The laws of motion are in general different in form for different frames of reference. When an arbitrary frame of reference is chosen, it may happen that the laws governing even very simple phenomena become very complex. The problem naturally arises of finding a frame of reference in which the laws of mechanics take their simplest form.

LL1/4-the-lagrangian-for-a-free-particle.md

@@ -1,5 +1,5 @@
 ---
-title: The Lagrangian for a free particle
+title: 4. The Lagrangian for a free particle
 ---
 
 Let us now go on to determine the form of the Lagrangian, and consider first of all the simplest case, that of the free motion of a particle relative to an inertial frame of reference. As we have already seen, the Lagrangian in this case can depend only on the square of the velocity. To discover the form of this dependence, we make use of Galileo's relativity principle. If an inertial frame $K$ is moving with an infinitesimal velocity $\v{\epsilon}$ relative to another inertial frame $K'$, then $\v{v}' = \v{v}+\v{\epsilon}$. Since the equations of motion must have the same form in every frame, the Lagrangian $L(v^2)$ must be converted by this transformation into a function $L'$ which differs from $L(v^2)$, if at all, only by the total time derivative of a function of co-ordinates and time (see the end of `LL1/2`).

LL1/5-the-lagrangian-for-a-system-of-particles.md

@@ -1,5 +1,5 @@
 ---
-title: The Lagrangian for a system of particles
+title: 5. The Lagrangian for a system of particles
 ---
 
 Let us now consider a system of particles which interact with one another but with no other bodies. This is called a closed system. It is found that the interaction between the particles can be described by adding to the Lagrangian `LL1/4.2` for non-interacting particles a certain function of the co-ordinates, which depends on the nature of the interaction[^1]. Denoting this function by $-U$, we have

LL1/6-energy.md

@@ -1,5 +1,5 @@
 ---
-title: Energy
+title: 6. Energy
 ---
 
 During the motion of a mechanical system, the $2s$ quantities $q_i$ and $\dot{q}_i$, $(i = 1, 2, s)$ which specify the state of the system vary with time. There exist, however, functions of these quantities whose values remain constant during the motion, and depend only on the initial conditions. Such functions are called integrals of the motion.

LL1/7-momentum.md

@@ -1,5 +1,5 @@
 ---
-title: Momentum
+title: 7. Momentum
 ---
 
 A second conservation law follows from the homogeneity of space. By virtue of this homogeneity, the mechanical properties of a closed system are unchanged by any parallel displacement of the entire system in space. Let us therefore consider an infinitesimal displacement $\v{\epsilon}$, and obtain the condition for the Lagrangian to remain unchanged.

LL1/8-centre-of-mass.md

@@ -1,5 +1,5 @@
 ---
-title: Centre of mass
+title: 8. Centre of mass
 ---
 
 The momentum of a closed mechanical system has different values in different (inertial) frames of reference. If a frame $K'$ moves with velocity $\v{V}$ relative to another frame $K$, then the velocities $\v{v}_a'$ and $\v{v}_a$ of the particles relative to the two frames are such that $\v{v}_a = \v{v}_a' + \v{V}$. The momenta $P$ and $P'$ in the two frames are therefore related by

LL1/9-angular-momentum.md

@@ -1,5 +1,5 @@
 ---
-title: Angular momentum
+title: 9. Angular momentum
 ---
 
 Let us now derive the conservation law which follows from the isotropy of space. This isotropy means that the mechanical properties of a closed system do not vary when it is rotated as a whole in any manner in space. Let us therefore consider an infinitesimal rotation of the system, and obtain the condition for the Lagrangian to remain unchanged.

LL1/toc.md

@@ -0,0 +1,69 @@
+---
+title: Mechanics
+subtitle: |
+    By L. D. Landau and E. M. Lifschitz  
+    Translated from the Russian by J. B. Sykes and J. S. Bell
+---
+I.  THE EQUATIONS OF MOTION
+1. [Generalized co-ordinates](1-generalized-co-ordinates.html)
+2. [The principle of least action](2-the-principle-of-least-action.html)
+3. [Galileos's relativity principle](3-galileos-relativity-principle.html)
+4. [The Lagrangian for a free particle](4-the-lagrangian-for-a-free-particle.html)
+5. [The Lagrangian for a system of particles](5-the-lagrangian-for-a-system-of-particles.html)
+II.  CONSERVATION LAWS
+6. [Energy](6-energy.html)
+7. [Momentum](7-momentum.html)
+8. [Centre of mass](8-centre-of-mass.html)
+9. [Angular momentum](9-angular-momentum.html)
+10. [Mechanical similarity](10-mechanical-similarity.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)
+15. [Kepler's problem]()
+IV.  COLLISION BETWEEN PARTICLES
+16. Disintegration of particles
+17. Elastic collisions
+18. Scattering
+19. Rutherford's formula
+20. Small-angle scattering
+V.  SMALL OSCILLATIONS
+21. Free oscillations in one dimension
+22. Forced oscillations
+23. Oscillations of systems with more than one degree of freedom
+24. Vibrations of molecules
+25. Damped oscillations
+26. Forced oscillations under friction
+27. Parametric resonance
+28. Anharmonic oscillations
+29. Resonance in non-linear oscillations
+30. Motion in a rapidly oscillating field
+VI.  MOTION OF A RIGID BODY
+31. Angular velocity
+32. The inertia tensor
+33. Angular momentum of a rigid body
+34. The equations of motion of a rigid body
+35. Eulerian angles
+36. Euler's equations
+37. The asymmetrical top
+38. Rigid bodies in contact
+39. Motion in a non-inertial frame of reference
+VII.  THE CANONICAL EQUATIONS
+40. Hamilton's equations
+41. The Routhian
+42. Poisson brackets
+43. The action as a function of the co-ordinates
+44. Maupertuis' principle
+45. Canonical transformations
+46. Liouville's theorem
+47. The Hamilton-Jacobi equation
+48. Separation of the variables
+49. Adiabatic invariants
+50. General properties of motion in `s` dimensions

Makefile

@@ -1,11 +1,9 @@
 MD = $(shell ls LL1/*.md)
-HTML = $(addprefix out/,$(MD:.md=.html)) out/LL1/index.html
+HTML = $(addprefix out/,$(MD:.md=.html))
+GOATCOUNTER = $(or $(PROD),false)
 
 all: $(HTML) | out
-	cp tools/style.css out
+	cp -r assets out
-
-out/LL1/index.html:
-	./tools/gen_index.sh > $@
 
 out/%.html: %.md
 	mkdir -p $(dir $@)
@@ -26,3 +24,10 @@ clean:
 	rm -rf out
 
 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/

assets/favicon.ico

@@ -0,0 +1,15 @@
+<?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>

assets/favicon.svg

@@ -0,0 +1,15 @@
+<?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"
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+
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+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>

assets/style.css

@@ -5,14 +5,10 @@
   max-width: 800px;
   text-align: justify;
 }
-h1 {
-  text-transform: capitalize;
-}
 hr { display: block; height: 1px;
     border: 0; border-top: 1px solid #000000;
     margin: 1em 0; padding: 0;
 }
-
 .tooltip {
   position: relative;
   display: inline-block;

tools/gen_index.sh

@@ -1,15 +0,0 @@
-#!/bin/bash
-
-book=LL1
-sections=$(find LL1 -name "*.md" | sort -h)
-
-echo "<ul>"
-for s in $sections; do
-    title=$(awk -F'title: ' '$0=$2' $s)
-    num=$(basename $s | tr -d -c 0-9)
-    href=$(basename $s | sed 's/.md/.html/')
-    cat <<-EOF
-<li><a href="$href">§$num. $title</a></li>
-EOF
-done
-echo "</ul>"

tools/header.html

@@ -1,3 +0,0 @@
-<link rel="stylesheet" type="text/css" href="colors.css">
-<style>
-</style>

tools/paragraph-template.html

@@ -1,17 +1,20 @@
 <!DOCTYPE html>
 <html xmlns="http://www.w3.org/1999/xhtml" lang="$lang$" xml:lang="$lang$"$if(dir)$ dir="$dir$"$endif$>
 
-<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="../style.css">
 <head>
-  <a href="index.html">index</a>
+  <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">
+  <a href="toc.html">index</a>
   <meta charset="utf-8" />
+  $if(goatcounter)$
+  <script data-goatcounter="https://llcotp.goatcounter.com/count"
+        async src="//gc.zgo.at/count.js">
+  </script>
+  $endif$
   <meta name="generator" content="pandoc" />
   <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
   <title>$if(title-prefix)$$title-prefix$ – $endif$$pagetitle$</title>
-  <style>
-    $styles.html()$
-  </style>
 $for(css)$
   <link rel="stylesheet" href="$css$" />
 $endfor$
@@ -26,9 +29,14 @@ $endfor$
 
 $if(title)$
 <header id="title-block-header">
-<h1 class="title"; style="text-align:left;float:left">$title$
+<h1 class="title">$title$
 </h1>
 
+$if(subtitle)$
+<h3>$subtitle$
+</h3>
+
+$endif$
 </header>
 <hr style="clear:both;"/>
 $endif$
@@ -53,9 +61,10 @@ $for(sources)$
 $endfor$
 </ul>
 $endif$
-<hr>
-<a href=>previous par</a>
-<a href=>next par</a>
 </body>
+<hr>
+<footer>
+  <a href="https://github.com/jzck/llcotp">source code</a>
+</footer>
 </div>
 </html>

tools/pf-filter.py

@@ -25,7 +25,7 @@ def action(elem, doc):
             book, eq = elem.text.split('/')
             html = f'''
             <span class="tooltip" class="math inline">
-            \(({eq})\)
+            ({eq})
             <span class="tooltiptext">{tooltiptext}</span>
             </span>
             '''