-
Notifications
You must be signed in to change notification settings - Fork 1
/
solutions.tex
392 lines (342 loc) · 11.7 KB
/
solutions.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
\documentclass[a4paper]{article}
% TODO: improve solutions.tex exercice title formatting
\author{M. Bivert}
\title{The Theoretical Minimum \\
{\Large Classical Mechanics - Solutions} \\
{\footnotesize Last version: %
\href{https://tales.mbivert.com/on-the-theoretical-minimum-solutions/}%
{tales.mbivert.com/on-the-theoretical-minimum-solutions/} or %
\href{https://github.com/mbivert/ttm}{github.com/mbivert/ttm}}
}
\usepackage{subfiles}
\usepackage[margin=1in]{geometry}
\usepackage{hyperref}
\usepackage[utf8]{inputenc}
% \noindent everywhere
\setlength\parindent{0pt}
\usepackage{mathtools}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{bm}
\usepackage{amssymb}
\usepackage{stmaryrd}
\usepackage{float} % \begin{figure}[H]
\usepackage{titlesec}
\usepackage{titletoc}
\usepackage{tikz}
\usetikzlibrary{snakes,calc,patterns,angles,quotes,decorations.pathmorphing,math,decorations.pathreplacing,automata,arrows.meta,positioning,external}
\tikzexternalize[prefix=tikz/]
\usepackage{tikz-3dplot}
% TODO: can't we use pgfplots all the time over tikz, or
% the reverse?
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepackage{longtable,array}
\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
% See I01E01.tex:/verbatimtab
\usepackage{moreverb}
\newtheorem{exercise}{Exercise}
\newtheorem{axiom}{Axiom}
\newtheorem{definition}{Definition}
\newtheorem{remark}{Remark}
\newtheorem{theorem}{Theorem}
\newcommand{\hr}{\noindent\rule{\textwidth}{0.4pt} \\}
\newcommand{\hrr}{\begin{center}\noindent\rule{0.5\textwidth}{0.4pt} \\\end{center}}
\newcommand{\bra}[1]{\langle #1|}
\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\braket}[2]{\left\langle #1 \middle| #2 \right\rangle}
\newcommand\omicron{o}
\newcommand\Omicron{O}
\newcommand{\uvec}[1]{\bm{\hat{#1}}}
\newcommand{\norm}[1]{\lVert #1 \rVert}
\newcommand{\abs}[1]{\lvert #1 \rvert}
\newcommand{\epsdelta}[6]{%
(\forall #1 > 0),\,(\exists #2 \in \mathbb{R}),\,%
(\forall #4 \in \mathbb{R}) \left(\abs{#4-#3} < #2\right)%
\Rightarrow \left(\abs{#5 - #6} < #1\right)
}
% \title for subfiles
\newcommand{\xtitle}{\title{The Theoretical Minimum \\
{\Large Classical Mechanics - Solutions} \\
{\large \jobname} \\
{\footnotesize Last version: %
\href{https://tales.mbivert.com/on-the-theoretical-minimum-solutions/}%
{tales.mbivert.com/on-the-theoretical-minimum-solutions/} or %
\href{https://github.com/mbivert/ttm}{github.com/mbivert/ttm}}
}}
% Essentially, https://tex.stackexchange.com/a/329928
\newcounter{interludes}
\renewcommand{\theinterludes}{\arabic{interludes}}
\newcommand{\interlude}[1]{\refstepcounter{interludes}
\section*{Interlude \theinterludes: #1}
\addcontentsline{toc}{section}{Interlude \theinterludes: #1}
}
\newcounter{subinterludes}
\renewcommand{\thesubinterludes}{\arabic{subinterludes}}
\newcommand{\subinterlude}[1]{\refstepcounter{subinterludes}
\subsection*{#1}
\addcontentsline{toc}{subsection}{#1}
}
\newcounter{subsubinterludes}
\renewcommand{\thesubsubinterludes}{\arabic{subsubinterludes}}
\newcommand{\subsubinterlude}[1]{\refstepcounter{subsubinterludes}
\subsubsection*{#1}
\addcontentsline{toc}{subsubsection}{#1}
}
% Awful way to alter both the ToC and *section formats.
\newcounter{lectures}
\renewcommand{\thelectures}{\arabic{lectures}}
\newcommand{\lecture}[1]{\refstepcounter{lectures}
\section*{Lecture \thelectures: #1}
\addcontentsline{toc}{section}{Lecture \thelectures: #1}
}
\newcounter{sublectures}
\renewcommand{\thesublectures}{\arabic{sublectures}}
\newcommand{\sublecture}[1]{\refstepcounter{sublectures}
\subsection*{#1}
\addcontentsline{toc}{subsection}{#1}
}
\newcounter{subsublectures}
\renewcommand{\thesubsublectures}{\arabic{subsublectures}}
\newcommand{\subsublecture}[1]{\refstepcounter{subsublectures}
\subsubsection*{#1}
\addcontentsline{toc}{subsubsection}{#1}
}
% ~ x/lecture/c/xappendix/, awful but that'll do
\newcounter{xappendixs}
\renewcommand{\thexappendixs}{\arabic{xappendixs}}
\newcommand{\xappendix}[1]{\refstepcounter{xappendixs}
\section*{Appendix \thexappendixs: #1}
\addcontentsline{toc}{section}{Appendix \thexappendixs: #1}
}
\newcounter{subxappendixs}
\renewcommand{\thesubxappendixs}{\arabic{subxappendixs}}
\newcommand{\subxappendix}[1]{\refstepcounter{subxappendixs}
\subsection*{#1}
\addcontentsline{toc}{subsection}{#1}
}
\newcounter{subsubxappendixs}
\renewcommand{\thesubsubxappendixs}{\arabic{subsubxappendixs}}
\newcommand{\subsubxappendix}[1]{\refstepcounter{subsubxappendixs}
\subsubsection*{#1}
\addcontentsline{toc}{subsubsection}{#1}
}
\begin{document}
\maketitle
\begin{abstract}
Below are solution proposals to the exercises of
\textit{The Theoretical Minimum - Classical Mechanics}, written
by Leonard Susskind and George Hrabovsky. An effort has been
so as to recall from the book all the referenced equations,
and to be rather verbose regarding mathematical details,
in line with the general tone of the series.
\end{abstract}
\tableofcontents
% --------------------------------------------------------
\lecture{The Nature of Classical Physics}
\sublecture{What Is Classical Physics?}
\subsublecture{Exercise 1/3}
\subfile{L01E01.tex}
\subsublecture{Exercise 2/3}
\subfile{L01E02.tex}
\sublecture{Rules That Are Not Allowed: The Minus-First Law}
\sublecture{Dynamical Systems with an Infinite Number of States}
\subsublecture{Exercise 3/3}
\subfile{L01E03.tex}
\sublecture{Cycles and Conservation Laws}
\sublecture{The Limits of Precision}
% --------------------------------------------------------
\interlude{Spaces, Trigonometry, and Vectors}
\subinterlude{Coordinates}
\subsubinterlude{Exercise 1/6}
\subfile{I01E01.tex}
\subinterlude{Trigonometry}
\subinterlude{Vectors}
\subsubinterlude{Exercise 2/6}
\subfile{I01E02.tex}
\subsubinterlude{Exercise 3/6}
\subfile{I01E03.tex}
\subsubinterlude{Exercise 4/6}
\subfile{I01E04.tex}
\subsubinterlude{Exercise 5/6}
\subfile{I01E05.tex}
\subsubinterlude{Exercise 6/6}
\subfile{I01E06.tex}
% --------------------------------------------------------
\lecture{Motion}
\sublecture{Mathematical Interlude: Differential Calculus}
\subsublecture{Exercise 1/8}
\subfile{L02E01.tex}
\subsublecture{Exercise 2/8}
\subfile{L02E02.tex}
\subsublecture{Exercise 3/8}
\subfile{L02E03.tex}
\subsublecture{Exercise 4/8}
\subfile{L02E04.tex}
\subsublecture{Exercise 5/8}
\subfile{L02E05.tex}
\sublecture{Particle Motion}
\sublecture{Example of Motion}
\subsublecture{Exercise 6/8}
\subfile{L02E06.tex}
\subsublecture{Exercise 7/8}
\subfile{L02E07.tex}
\subsublecture{Exercise 8/8}
\subfile{L02E08.tex}
% --------------------------------------------------------
\interlude{Integral Calculus}
\subinterlude{Integral Calculus}
\subsubinterlude{Exercise 1/4}
\subfile{I02E01.tex}
\subsubinterlude{Exercise 2/4}
\subfile{I02E02.tex}
\subsubinterlude{Exercise 3/4}
\subfile{I02E03.tex}
\subinterlude{Integration by Parts}
\subsubinterlude{Exercise 4/4}
\subfile{I02E04.tex}
% --------------------------------------------------------
\lecture{Dynamics}
\sublecture{Aristotle's Law of Motion}
\subsublecture{Exercise 1/4}
\subfile{L03E01.tex}
\sublecture{Mass, Acceleration and Force}
\sublecture{Some Simple Examples of Solving Newton's Equations}
\subsublecture{Exercise 2/4}
\subfile{L03E02.tex}
\subsublecture{Exercise 3/4}
\subfile{L03E03.tex}
\subsublecture{Exercise 4/4}
\subfile{L03E04.tex}
% --------------------------------------------------------
\interlude{Partial Differentiation}
\subinterlude{Partial Derivatives}
\subsubinterlude{Exercise 1/2}
\subfile{I03E01.tex}
\subinterlude{Stationary Points and Minimizing Functions}
\subinterlude{Stationary Points in Higher Dimensions}
\subsubinterlude{Exercise 2/2}
%\subfile{I03E02.tex}
% --------------------------------------------------------
\lecture{Systems of More Than One Particle}
\sublecture{Systems of Particles}
\sublecture{The Space of States of a System of Particles}
\sublecture{Momentum and Phase Space}
\sublecture{Action, Reaction, and the Conservation of Momentum}
% --------------------------------------------------------
\lecture{Energy}
\sublecture{Force and Potential Energy}
\subsublecture{Exercise 1/3}
\subfile{L05E01.tex}
\sublecture{More than one dimension}
\subsublecture{Exercise 2/3}
\subfile{L05E02.tex}
\subsublecture{Exercise 3/3}
\subfile{L05E03.tex}
% --------------------------------------------------------
\lecture{The Principle of Least Action}
\sublecture{The Transition to Advanced Mechanics}
\sublecture{Action and the Lagrangian}
\sublecture{Derivation of the Euler-Lagrange Equation}
\subsublecture{Exercise 1/6}
\subfile{L06E01.tex}
\sublecture{More Particles and More dimensions}
\subsublecture{Exercise 2/6}
\subfile{L06E02.tex}
\sublecture{What's Good about Least Action?}
\subsublecture{Exercise 3/6}
\subfile{L06E03.tex}
\subsublecture{Exercise 4/6}
\subfile{L06E04.tex}
\sublecture{Generalized Coordinates and Momenta}
\subsublecture{Exercise 5/6}
%\subfile{L06E05.tex}
\sublecture{Cyclic coordinates}
\subsublecture{Exercise 6/6}
\subfile{L06E06.tex}
% --------------------------------------------------------
\lecture{Symmetries and Conservation Laws}
\sublecture{Preliminaries}
\subsublecture{Exercise 1/7}
\subfile{L07E01.tex}
\subsublecture{Exercise 2/7}
\subfile{L07E02.tex}
\sublecture{Examples of symmetries}
\subsublecture{Exercise 3/7}
\subfile{L07E03.tex}
\subsublecture{Exercise 4/7}
\subfile{L07E04.tex}
\sublecture{Back to examples}
\subsublecture{Exercise 5/7}
%\subfile{L07E05.tex}
\subsublecture{Exercise 6/7}
%\subfile{L07E06.tex}
\subsublecture{Exercise 7/7}
%\subfile{L07E07.tex}
% --------------------------------------------------------
\lecture{Hamiltonian Mechanics and Time-Translation Invariance}
\sublecture{Time-Translation Symmetry}
\sublecture{Energy Conservation}
\sublecture{Phase Space and Hamilton's Equations}
\sublecture{The Harmonic Oscillator Hamiltonian}
\subsublecture{Exercise 1/2}
\subfile{L08E01.tex}
\subsublecture{Exercise 2/2}
\subfile{L08E02.tex}
\sublecture{Derivation of Hamilton's Equations}
% --------------------------------------------------------
\lecture{The Phase Space Fluid and the Gibbs-Liouville Theorem}
\sublecture{The Phase Space Fluid}
\sublecture{A Quick Reminder}
\sublecture{Flow and Divergence}
\sublecture{Liouville's Theorem}
\sublecture{Poisson Brackets}
% --------------------------------------------------------
\lecture{Poisson Brackets, Angular Momentum, and Symmetries}
\sublecture{An Axiomatic Formulation of Mechanics}
\subsublecture{Exercise 1/3}
\subfile{L10E01.tex}
\subsublecture{Exercise 2/3}
\subfile{L10E02.tex}
\sublecture{Angular Momentum}
\subsublecture{Exercise 3/3}
\subfile{L10E03.tex}
\sublecture{Mathematical Interlude - The Levi-Civita Symbol}
\sublecture{Back to Angular Momentum}
\sublecture{Rotors and Precession}
\sublecture{Symmetry and Conservation}
% --------------------------------------------------------
\lecture{Electric and Magnetic Forces}
\sublecture{Vector Fields}
\sublecture{Mathematical Interlude: Del}
\subsublecture{Exercise 1/5}
\subfile{L11E01.tex}
\subsublecture{Exercise 2/5}
\subfile{L11E02.tex}
\sublecture{Magnetic Fields}
\subsublecture{Exercise 3/5}
\subfile{L11E03.tex}
\sublecture{The Force on a Charged Particle}
\sublecture{The Lagrangian}
\sublecture{Equations of Motion}
\sublecture{The Hamiltonian}
\subsublecture{Exercise 4/5}
\subfile{L11E04.tex}
\sublecture{Motion in a Uniform Magnetic Field}
\subsublecture{Exercise 5/5}
\subfile{L11E05.tex}
\sublecture{Gauge Invariance}
\sublecture{Good Bye for Now}
% --------------------------------------------------------
\xappendix{Central Forces and Planetary Orbits}
\subxappendix{The Central Force of Gravity}
\subxappendix{Gravitational Potential Energy}
\subxappendix{The Earth Moves in a Plane}
\subxappendix{Polar Coordinates}
\subxappendix{Equations of Motion}
\subxappendix{Effective Potential Energy Diagrams}
\subxappendix{Kepler's Laws}
\subsubxappendix{Exercise 1/1}
%\subfile{A01E01.tex}
\end{document}