Table Of ContentLAGRANGIAN & HAMILTONIAN DYNAMICS
Lagrangian & Hamiltonian Dynamics
Peter Mann
University of St Andrews
1
3
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ISBN 978–0–19–882238–7 (pbk.)
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For my beautiful daughter Hallie.
Contents
Preface
xiii
PART I
NEWTONIAN MECHANICS
1
Newton’s Three Laws
3
1.1
Phase Space
7
1.2
Systems of Particles
8
1.3
The N-body Problem
11
Chapter summary
13
2
Energy and Work
15
Chapter summary
22
3
Introductory Rotational Dynamics
24
Chapter summary
33
4
The Harmonic Oscillator
34
Chapter summary
38
5
Wave Mechanics & Elements of Mathematical Physics
44
PART II
LAGRANGIAN MECHANICS
6
Coordinates & Constraints
55
Chapter summary
59
7
The Stationary Action Principle
61
7.1
The Inverse Problem
70
7.2
Higher-Order Theories & the Ostrogradsky Equation
72
7.3
The Second Variation
73
7.4
Functions & Functionals
74
7.5
Boundary Conditions
76
7.6
Variations
78
7.7
Weierstrass-Erdmann Conditions for Broken Extremals
79
7.8
Hamilton-Suslov Principle
79
Chapter summary
80
8
Constrained Lagrangian Mechanics
89
8.1
Holonomic Constraints
89
viii
Contents
8.2
Non-Holonomic Constraints
93
Chapter summary
96
9
Point Transformations in Lagrangian Mechanics
100
Chapter summary
103
10 The Jacobi Energy Function
107
Chapter summary
112
11 Symmetries & Lagrangian-Hamilton-Jacobi Theory
115
11.1 Noether’s Theorem
115
11.2 Gauge Theory
120
11.3 Isotropic Symmetries
122
11.4 Caratheodory-Hamilton-Jacobi theory
123
Chapter summary
124
12 Near-Equilibrium Oscillations
130
12.1 Normal Modes
137
Chapter summary
140
13 Virtual Work & d’Alembert’s Principle
147
13.1 Gauss’s Least Constraint & Jourdain’s Principle
153
13.2 The Gibbs-Appell Equations
156
Chapter summary
158
PART III
CANONICAL MECHANICS
14 The Hamiltonian & Phase Space
167
Chapter summary
172
15 Hamilton’s Principle in Phase Space
174
Chapter summary
178
16 Hamilton’s Equations & Routhian Reduction
179
16.1 Phase Space Conservation Laws
181
16.2 Routhian Mechanics
183
17 Poisson Brackets & Angular Momentum
190
17.1 Poisson Brackets & Angular Momenta
195
17.2 Poisson Brackets & Symmetries
197
Chapter summary
200
18 Canonical & Gauge Transformations
202
18.1 Canonical Transformations I
202
18.2 Canonical Transformations II
206
Contents
ix
18.3 Infinitesimal Canonical Transformations
211
Chapter summary
214
19 Hamilton-Jacobi Theory
217
19.1 Hamilton-Jacobi Theory I
217
19.2 Hamilton-Jacobi Theory II
224
Chapter summary
229
20 Liouville’s Theorem & Classical Statistical Mechanics
237
20.1 Liouville’s Theorem & the Classical Propagator
237
20.2 Koopman-von Neumann Theory
244
20.3 Classical Statistical Mechanics
246
20.4 Symplectic Integrators
255
Chapter summary
259
21 Constrained Hamiltonian Dynamics
267
Chapter summary
274
22 Autonomous Geometrical Mechanics
277
22.1 A Coordinate-Free Picture
284
22.2 Poisson Manifolds & Symplectic Reduction
291
22.3 Geometrical Lagrangian Mechanics
296
22.4 Elements of Constrained Geometry
300
Chapter summary
303
23 The Structure of Phase Space
309
23.1 Time-Dependent Geometrical Mechanics
313
23.2 Picturing Phase Space
319
Chapter summary
322
24 Near-Integrable Systems
325
24.1 Canonical Perturbation Theory
325
24.2 KAM Theory & Elements of Chaos
333
PART IV
CLASSICAL FIELD THEORY
25 Lagrangian Field Theory
345
Chapter summary
350
26 Hamiltonian Field Theory
353
27 Classical Electromagnetism
357
Chapter summary
365
x
Contents
28 Noether’s Theorem for Fields
369
Chapter summary
376
29 Classical Path-Integrals
385
29.1 Configuration Space Integrals
385
29.2 Phase Space Integrals
386
PART V
PRELIMINARY MATHEMATICS
30 The (Not So?) Basics
397
31 Matrices
400
32 Partial Differentiation
406
33 Legendre Transforms
419
34 Vector Calculus
422
35 Differential Equations
437
36 Calculus of Variations
443
PART VI
ADVANCED MATHEMATICS
37 Linear Algebra
453
38 Differential Geometry
460
PART VII
EXAM-STYLE QUESTIONS
Appendix A
Noether’s Theorem Explored
487
Appendix B
The Action Principle Explored
491
B.1
Geodesics
491
Appendix C
Useful Relations
494
Appendix D
Poisson & Nambu Brackets Explored
496
D.1
Symplectic Notation & Nambu Brackets
497
Appendix E
Canonical Transformations Explored
502
Appendix F
Action-Angle Variables Explored
506
Appendix G
Statistical Mechanics Explored
511
G.1
The Boltzmann Factor
511
G.2
Fluctuations
512
Contents
xi
Appendix H
Biographies
514
H.1
Sir Isaac Newton
514
H.2
Leonhard Euler
515
H.3
Jean d’Alembert
516
H.4
Joseph-Louis Lagrange
517
H.5
Carl Gustav Jacobi
519
H.6
Sir William Hamilton
520
H.7
Siméon Denis Poisson
522
H.8
Amalie Emmy Noether
522
H.9
Ludwig Eduard Boltzmann
524
H.10 Edward Routh
525
H.11 Hendrika van Leeuwen
526
Bibliography
527
Index
533
Preface
The purpose of this book is to introduce and explore the subject of Lagrangian and
Hamiltonian dynamics to science students within a relaxed and self-contained setting
for those unacquainted with mathematics or university-level physics. Lagrangian and
Hamiltonian dynamics is the continuation of Newton’s classical physics into new for-
malisms, each highlighting novel aspects of mechanics that gradually build in complex-
ity to form the basis for almost all of theoretical physics. Lagrangian and Hamiltonian
dynamics also acts as a gateway to more abstract concepts routed in differential geom-
etry and field theories and can be used to introduce these subject areas to newcomers.
In this book we journey in a self-contained manner from the very basics, through the
fundamentals and onwards to the cutting edge of the subject at the forefront of research.
Along the way, the reader is supported by all the necessary background mathematics,
fully worked examples and thoughtful and vibrant illustrations, as well as an informal
narrative and numerous fresh, modern and interdisciplinary applications. For example,
the subject is rarely discussed within the scope of chemistry, biology or medicine, despite
numerous applications and an absolute relevance.
The book contains a very detailed and explicit account of Lagrangian and Hamil-
tonian dynamics that ventures above and beyond most undergraduate courses in the
UK. Alongside this, there are some unusual topics for a classical mechanics textbook.
The most notable examples include the “classical wavefunction” Koopman-von Neu-
mann theory, classical density functional theories, the “vakonomic” variational princi-
ple for non-holonomic constraints, the Gibbs-Appell equations, classical path integrals,
Nambu brackets, Lagrangian-Hamilton-Jacobi theory, Dirac brackets and the full fram-
ing of mechanics in the language of differential geometry, alongside many more unique
features!
The book features many fully worked examples to complement the core material;
these can be attempted as exercises beforehand. It is here that many modern applica-
tions of mechanics are investigated, such as protein folding, atomic force microscopy,
medical imaging, cell membrane modelling and surface adsorbate analysis, to name
but a few! Key equations are highlighted throughout the text, and colour is used in
derivations that could otherwise be hard to follow. At the back of the book there are
vast mathematical chapters that cover all the material that is required to understand
classical mechanics, so that the book is a self-contained reference.
The motivation for writing this text was generated during my undergraduate de-
gree in chemistry. Often, I would try to research further information on my quantum
xiv
Preface
mechanics courses, only to be inundated with insurmountable walls of equations and
complex ideas which resulted in many opened Wikipedia tabs! Out of frustration, I be-
gan compiling a set of notes to aid my understanding; naturally, it got completely out
of hand and resulted in the current work! I found myself referring back to my own notes
in order to understand new concepts and, very quickly, I fell in love with mechanics,
appreciating its beauty more and more with each new formulation I studied. It is hence
for those without formal training but who wish to understand more about physics that
I wrote this book. It is for this reason that every mathematical step is fully detailed,
and plenty of diagrams and examples have been packed into the book to help along the
way! A brief biography of the characters who developed the ideas we present is given
at the back of the book, for those interested in the history of physics.
I don’t think I could pick a “favourite equation” out of the book, or even a favourite
formulation - each bring their own elegance and unique features which stand out among
the rest! Koopman-von Neumann theory is extremely interesting and is useful for prob-
ing the structural features of mechanics but, on the other hand, d’Alembert’s principle
and the Gibbs-Appell equations are simply beautiful! I happen to think however that
Newton and Euler are possibly the two cleverest individuals in the history of natural
philosophy, as, without them, we would have no mechanics!
The book can be used in many ways and for different audiences; for an interested
individual, it can provide further information on a particular topic; alternatively, it
can serve as a support for undergraduate courses in classical mechanics. There are per-
haps three types of courses this book would suit: (i) courses on Newtonian mechanics
and classical mathematical physics; (ii) introductory Lagrangian and Hamiltonian dy-
namics courses at the undergraduate level; (iii) advanced courses on mechanics at the
postgraduate level.
I would like to thank Duncan Stewart for his illustrations, which far exceeded my
expectations and crude drawings. I would like to thank Elizabeth Farrell for correcting
my ferocious lack of punctuation and terrible grammar. I would also like to thank
Ania Wronski and Lydia Shinoj for their help in the final stages of the book. Thank
Renald Schaub and Collin Bleak for their kind encouragements and John Mitchell,
Bernd Braunecker and Jonathan Keeling for their supportive feedback. In addition I
would like to thank my tutors at Liverpool College and my lecturers at the University
of St Andrews, whose passion and enthusiasm for science I have inherited. I would like
to thank all my friends for their encouragement and for always being up for a kick
about with the football at any spontaneous moment—in particular Giorgio and Josh.
Above all, I would like to thank my parents and my beautiful wife, whose love, support
and encouragements, from the opening lines to the final edits, made this book a reality!
So here we go—I hope you enjoy reading it as much as I enjoyed writing it!
Peter Mann
