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Oxford Graduate Texts in Mathematics
R. Cohen
Series Editors
S. K. Donaldson S. Hildebrandt
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OXFORD GRADUATE TEXTS IN MATHEMATICS
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Keith Hannabuss: An introduction to quantum theory
Reinhold Meise and Dietmar Vogt: Introduction to functional analysis
James G. Oxley: Matroid theory
N. J. Hitchin, G. B. Segal, and R. S. Ward: Integrable systems: twistors,
loop groups, and Riemann surfaces
Wulf Rossmann: Lie groups: An introduction through linear groups
Qing Liu: Algebraic geometry and arithmetic curves
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Maciej Dunajski: Solitons, Instantons, and Twistors
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Solitons, Instantons, and
Twistors
Maciej Dunajski
Department of Applied Mathematics and Theoretical
Physics, University of Cambridge
1
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3
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ISBN 978–0–19–857062–2 (Hbk.)
978–0–19–857063–9 (Pbk.)
1 3 5 7 9 10 8 6 4 2
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Preface
This book grew out of lecture courses I have given to mathematics Part II, Part
III, and graduate students in Cambridge between the years 2003 and 2008.
The first four chapters could form a basis of a one-term lecture course on
integrable systems covering the Arnold–Liouville theorem, inverse scattering
transform, Hamiltonian methods in soliton theory, and Lie point symmetries.
The additional, more advanced topics are covered in Chapters 7 and 8. They
include the anti-self-dual Yang–Mills equations, their symmetry reductions,
and twistor methods. Chapters 5, 6, and 9 provide material for an advanced
course on field theory where particular emphasis is paid to non-perturbative
solutions to classical field equations. We shall discuss scalar kinks, sigmamodel lumps, non-abelian magnetic monopoles in R3 , instanton solutions
to pure Yang–Mills equations in R4 , and finally gravitational instantons.
Although the material is entirely ‘classical’, the motivation comes from quantum field theories, including gauge theories, where it is necessary to consider
solutions of the non-linear field equations that are topologically distinct from
the vacuum. Chapter 10 contains a discussion of the anti-self-dual conformal
structures, some of which have not been presented in literature before. This
chapter links with the rest of the book as the anti-self-dual conformal structures provide a unifying framework for studying the dispersionless integrable
systems. There are three appendices. The first two provide a mathematical
background in topology of manifolds and Lie groups (Appendix A), and
complex analysis required by twistor theory (Appendix B). Appendix C is selfcontained and can form a basis of a one-term lecture course on overdetermined
partial differential equations. This appendix gives an elementary introduction
to the subject known as the exterior differential system.
Although the term soliton – a localized non-singular lump of energy –
plays a central role in the whole book, its precise meaning changes as the
reader progresses through the chapters. In Chapters 1, 2, 3, and 8 solitons
arise as solutions to completely integrable equations. They are time-dependent
localized waves which scatter without emitting radiation, and owe their stability to the existence of infinitely many dynamically conserved currents. In
Chapters 5 and 6 the solitons are topological – they are characterized by
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vi
Preface
discrete homotopy invariants (usually in the form of Chern numbers) which
are conserved as the continuous field configurations have finite energy. Finally
in Chapter 9 the gravitational solitons arise as lifts of Riemannian gravitational
instantons to higher dimensions.
The book should be of use to advanced undergraduate and research students, as well as experts in soliton theory who want to broaden their techniques. It is aimed at both mathematicians and those physicists who are willing
to go beyond perturbation theory. The revived interest in twistor theory in
recent years can be largely attributed to Witten’s twistor-string theory [185]. It
is hoped that those researchers who come to twistor strings with the string
theory or quantum field theory (QFT) background will find this book an
accessible introduction to twistor theory.
There are some excellent text books which treat the material presented
here in great depth. Readers should consult [122] for inverse scattering transform, [124] for the symmetry methods, and [114] for topological solitons.
The twistor approach to integrability is a subject of the monograph [118],
while [83, 132, 175] are books on twistors which concentrate on aspects of
the theory other than integrability. The full treatment of exterior differential
systems can be found in [23].
The twistor approach to integrability used in the second half of the book
has been developed over the last thirty years by the Oxford school of Sir Roger
Penrose with a particular input from Richard Ward. I am grateful to Sir Roger
for sharing his inspirational ideas with the rest of us. His original motivation
was to unify general relativity and quantum mechanics in a non-local theory
based on complex numbers. The application of twistor theory to integrability
has been an unexpected spin-off from the twistor programme.
While preparing the manuscript I have benefited from many valuable discussions with my colleagues, collaborators, and research students. In particular I
would like to thank Robert Bryant, David Calderbank, Mike Eastwood, Jenya
Ferapontov, Gary Gibbons, Sean Hartnoll, Nigel Hitchin, Marcin Ka´zmierczak, Piotr Kosinski, Nick Manton, Lionel Mason, Vladimir Matveev, Pawe l
Nurowski, Roger Penrose, Prim Plansangkate, Maciej Przanowski, George
Sparling, David Stuart, Paul Tod, Simon West, and Nick Woodhouse. I am
especially grateful to Paul Tod for carefully reading the manuscript.
Finally, I thank my wife Asia and my sons Adam and Nico not least for
making me miss several submission deadlines. I dedicate this book to the three
of them with gratitude.
Cambridge
January 2009
Maciej Dunajski
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Contents
List of Figures
xii
List of Abbreviations
xiii
1
2
3
Integrability in classical mechanics
1
1.1 Hamiltonian formalism
1.2 Integrability and action–angle variables
1.3 Poisson structures
1
4
14
Soliton equations and the inverse scattering transform
20
2.1 The history of two examples
2.1.1 A physical derivation of KdV
2.1.2 Bäcklund transformations for the Sine-Gordon equation
2.2 Inverse scattering transform for KdV
2.2.1 Direct scattering
2.2.2 Properties of the scattering data
2.2.3 Inverse scattering
2.2.4 Lax formulation
2.2.5 Evolution of the scattering data
2.3 Reflectionless potentials and solitons
2.3.1 One-soliton solution
2.3.2 N-soliton solution
2.3.3 Two-soliton asymptotics
20
21
24
25
28
29
30
31
32
33
34
35
36
Hamiltonian formalism and zero-curvature representation
43
3.1 First integrals
3.2 Hamiltonian formalism
3.2.1 Bi-Hamiltonian systems
43
46
46
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Contents
4
5
6
3.3 Zero-curvature representation
3.3.1 Riemann–Hilbert problem
3.3.2 Dressing method
3.3.3 From Lax representation to zero curvature
3.4 Hierarchies and finite-gap solutions
48
50
52
54
56
Lie symmetries and reductions
64
4.1 Lie groups and Lie algebras
4.2 Vector fields and one-parameter groups of
transformations
4.3 Symmetries of differential equations
4.3.1 How to find symmetries
4.3.2 Prolongation formulae
4.4 Painlevé equations
4.4.1 Painlevé test
64
67
71
74
75
78
82
Lagrangian formalism and field theory
85
5.1 A variational principle
5.1.1 Legendre transform
5.1.2 Symplectic structures
5.1.3 Solution space
5.2 Field theory
5.2.1 Solution space and the geodesic
approximation
5.3 Scalar kinks
5.3.1 Topology and Bogomolny equations
5.3.2 Higher dimensions and a scaling argument
5.3.3 Homotopy in field theory
5.4 Sigma model lumps
85
87
88
89
90
92
93
96
98
99
100
Gauge field theory
105
6.1 Gauge potential and Higgs field
6.1.1 Scaling argument
6.1.2 Principal bundles
6.2 Dirac monopole and flux quantization
6.2.1 Hopf fibration
6.3 Non-abelian monopoles
6.3.1 Topology of monopoles
6.3.2 Bogomolny–Prasad–Sommerfeld (BPS) limit
106
108
109
110
112
114
115
116
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Contents
7
8
9
6.4 Yang–Mills equations and instantons
6.4.1 Chern and Chern–Simons forms
6.4.2 Minimal action solutions and the anti-self-duality
condition
6.4.3 Ansatz for ASD fields
6.4.4 Gradient flow and classical mechanics
119
120
Integrability of ASDYM and twistor theory
129
7.1 Lax pair
7.1.1 Geometric interpretation
7.2 Twistor correspondence
7.2.1 History and motivation
7.2.2 Spinor notation
7.2.3 Twistor space
7.2.4 Penrose–Ward correspondence
129
132
133
133
137
139
141
Symmetry reductions and the integrable chiral model
149
8.1 Reductions to integrable equations
8.2 Integrable chiral model
8.2.1 Soliton solutions
8.2.2 Lagrangian formulation
8.2.3 Energy quantization of time-dependent unitons
8.2.4 Moduli space dynamics
8.2.5 Mini-twistors
149
154
157
165
168
173
181
Gravitational instantons
191
9.1 Examples of gravitational instantons
9.2 Anti-self-duality in Riemannian geometry
9.2.1 Two-component spinors in Riemannian signature
9.3 Hyper-Kähler metrics
9.4 Multi-centred gravitational instantons
9.4.1 Belinskii–Gibbons–Page–Pope class
9.5 Other gravitational instantons
9.5.1 Compact gravitational instantons and K3
9.6 Einstein–Maxwell gravitational instantons
9.7 Kaluza–Klein monopoles
9.7.1 Kaluza–Klein solitons from Einstein–Maxwell
instantons
9.7.2 Solitons in higher dimensions
191
195
198
202
206
210
212
215
216
221
122
123
124
222
226
ix
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x
Contents
10 Anti-self-dual conformal structures
10.1 α-surfaces and anti-self-duality
10.2 Curvature restrictions and their Lax pairs
10.2.1 Hyper-Hermitian structures
10.2.2 ASD Kähler structures
10.2.3 Null-Kähler structures
10.2.4 ASD Einstein structures
10.2.5 Hyper-Kähler structures and heavenly equations
10.3 Symmetries
10.3.1 Einstein–Weyl geometry
10.3.2 Null symmetries and projective structures
10.3.3 Dispersionless integrable systems
10.4 ASD conformal structures in neutral signature
10.4.1 Conformal compactification
10.4.2 Curved examples
10.5 Twistor theory
10.5.1 Curvature restrictions
10.5.2 ASD Ricci-flat metrics
10.5.3 Twistor theory and symmetries
Appendix A: Manifolds and topology
A.1 Lie groups
A.2 Degree of a map and homotopy
A.2.1 Homotopy
A.2.2 Hermitian projectors
Appendix B: Complex analysis
B.1 Complex manifolds
B.2 Holomorphic vector bundles and their sections
ˇ
B.3 Cech
cohomology
B.3.1 Deformation theory
Appendix C: Overdetermined PDEs
C.1 Introduction
C.2 Exterior differential system and Frobenius theorem
C.3 Involutivity
229
230
231
232
234
236
237
238
246
246
253
256
262
263
263
265
270
272
283
287
290
294
296
298
300
301
303
307
308
310
310
314
320
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Contents
C.4 Prolongation
C.4.1 Differential invariants
C.5 Method of characteristics
C.6 Cartan–Kähler theorem
324
326
332
335
References
344
Index
355
xi
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List of Figures
1.1 Level surface
3
1.2 Branch cut for the Kepler integral
12
2.1 Sine-Gordon kink
25
2.2 Reflection and transmission
29
5.1 Multiple vacuum
94
5.2 Kink, N = 1
96
5.3 Anti-kink, N = −1
96
5.4 Kink–anti-kink pair, N = 0
96
7.1 Twistor correspondence
140
8.1 Suspension and reduced suspension
163
8.2 A geodesic joining two points
187
8.3 Blow-up of the vertex of the cone
188
10.1 Double fibration
267
10.2 Relationship between M, U, PT , and Z
284
A.1 Manifold
288
B.1 Splitting formula
308
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List of Abbreviations
ASDYM
Anti-self-dual Yang-Mills
AHS
Atiyah–Hitchin–Singer
ALE
Asymptotically locally Euclidean
ALF
Asymptotically locally flat
ASD
Anti-self-dual
BGPP
Belinskii–Gibbons–Page–Pope
BPS
Bogomolny–Prasad–Sommerfeld
CR
Cauchy–Riemann
DE
Differential equation
dKP
Dispersionless Kadomtsev–Petviashvili
EDS
Exterior differential system
EW
Einstein–Weyl
GLM
Gelfand–Levitan–Marchenko
IST
Inverse scattering transform
KdV
Korteweg–de Vries
LHC
Large hadron collider
LHS
Left-hand side
ODE
Ordinary differential equation
PDE
Partial differential equation
PP
Painlevé property
QM
Quantum mechanics
RHS
Right-hand side
SD
Self-dual
WZW
Wess–Zumino–Witten
YM
Yang–Mills
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Integrability in classical
mechanics
1
Integrable systems are non-linear differential equations (DEs) which ‘in principle’ can be solved analytically. This means that the solution can be reduced to
a finite number of algebraic operations and integrations. Such systems are very
rare – most non-linear DEs admit chaotic behaviour and no explicit solutions
can be written down. Integrable systems nevertheless lead to very interesting
mathematics ranging from differential geometry and complex analysis to quantum field theory and fluid dynamics. In this chapter we shall introduce the
integrability of ordinary differential equations (ODEs). This is a fairly clear
concept based on existence of sufficiently many well-behaved first integrals, or,
as a physicist would put it, constants of the motion.
1.1
Hamiltonian formalism
Motion of a system with n degrees of freedom is described by a trajectory in a
2n dimensional phase space M (locally think of an open set in R2n but globally
it can be a topologically non-trivial manifold – for example, a sphere or a
torus. See Appendix A) with local coordinates
( p j , q j ),
j = 1, 2, . . . , n.
The dynamical variables are functions f : M × R −→ R, so that f = f ( p, q, t)
where t is called ‘time’. Let f, g : M × R −→ R. Define the Poisson bracket of
f, g to be the function
n
{ f, g} :=
k=1
∂ f ∂g
∂ f ∂g
−
.
∂qk ∂ pk ∂ pk ∂qk
It satisfies
{ f, g} = −{g, f },
{ f, {g, h}} + {g, {h, f }} + {h, { f, g}} = 0.
(1.1.1)
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1 : Integrability in classical mechanics
These two properties are called the skew-symmetry and the Jacobi identity,
respectively. One says that two functions f, g are in involution if { f, g} = 0.
The coordinate functions ( p j , q j ) satisfy the canonical commutation relations
{ p j , pk} = 0,
{q j , qk} = 0, and {q j , pk} = δ jk.
Given a Hamiltonian H = H( p, q, t) (usually H( p, q)) the dynamics is determined by
∂f
df
=
+ { f, H},
dt
∂t
for any
f = f ( p, q, t).
Setting f = p j or f = q j yields Hamilton’s equations of motion
p˙ j = −
∂H
∂q j
and q˙ j =
∂H
.
∂ pj
(1.1.2)
The system (1.1.2) of 2n ODEs is deterministic in the sense that ( p j (t), q j (t))
are uniquely determined by 2n initial conditions ( p j (0), q j (0)). Equations
(1.1.2) also imply that volume elements in phase space are conserved. This
system is essentially equivalent to Newton’s equations of motion. The Hamiltonian formulation allows a more geometrical insight into classical mechanics.
It is also the starting point to quantization.
Definition 1.1.1 A function f = f ( p j , q j , t) which satisfies f˙ = 0 when equations (1.1.2) hold is called a first integral or a constant of motion. Equivalently,
f ( p(t), q(t), t) = const
if p(t), q(t) are solutions of (1.1.2).
In general the system (1.1.2) will be solvable if it admits ‘sufficiently many’
first integrals and the reduction of order can be applied. This is because any
first integral eliminates one equation.
r Example. Consider a system with one degree of freedom with M = R2 and
the Hamiltonian
H( p, q) =
1 2
p + V(q).
2
Hamilton’s equations (1.1.2) give
q˙ = p and p˙ = −
dV
.
dq
The Hamiltonian itself is a first integral as {H, H} = 0. Thus
1 2
p + V(q) = E
2
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1.1
Hamiltonian formalism
M
f(p, q)= constant
Figure 1.1
Level surface
where E is a constant called energy. Now
q˙ = p,
p = ± 2[E − V(q)]
and one integration gives a solution in the implicit form
t=±
dq
2[E − V(q)]
.
The explicit solution could be found if we can perform the integral on the
right-hand side (RHS) and invert the relation t = t(q) to find q(t). These two
steps are not always possible but nevertheless we would certainly regard this
system as integrable.
It is useful to adopt a more geometrical approach. Assume that a first integral
f does not explicitly depend on time, and that it defines a hypersurface
f ( p, q) = const in M (Figure.1.1). Two hypersurfaces corresponding to two
independent first integrals generically intersect in a surface of co-dimension 2
in M. In general the trajectory lies on a surface of dimension 2n − L where L is
the number of independent first integrals. If L = 2n − 1 this surface is a curve –
a solution to (1.1.2).
How may we find first integrals? Given two first integrals which do not
explicitly depend on time their Poisson bracket will also be a first integral if it
is not zero. This follows from the Jacobi identity and the fact all first integrals
Poisson commute with the Hamiltonian. More generally, the Noether theorem
gives some first integrals (this theorem relates the symmetries that Hamilton’s
equation (1.1.2) may possess, for example, time translation, rotations, etc., to
first integrals) but not enough. The difficulty with finding the first integrals has
deep significance. For assume we use some existence theorem for ODEs and
apply it to (1.1.2). Now solve the algebraic equations
qk = qk( p0 , q0 , t) and pk = pk( p0 , q0 , t),
3
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4
1 : Integrability in classical mechanics
for the initial conditions ( p0 , q0 ) thus giving
q0 k = q0 k( p, q, t) and p0 k = p0 k( p, q, t).
This gives 2n first integrals as obviously ( p0 , q0 ) are constants which we can
freely specify. One of these integrals determines the time parametrizations and
others could perhaps be used to construct the trajectory in the phase space.
However for some of the integrals the equation
f ( p, q) = const
may not define a ‘nice’ surface in the phase space. Instead it defines a pathological (at least from the applied mathematics point of view) set which densely
covers the phase space. Such integrals do not separate points in M.
One first integral – energy – always exist for Hamiltonian systems giving the
energy surface H( p, q) = E, but often it is the only first integral. Sufficiently
complicated, deterministic systems may behave according to the laws of thermodynamics: the probability that the system is contained in some element of
the energy surface is proportional to the normalized volume of this element.
This means that the time evolution covers uniformly the entire region of the
constant energy surface in the phase space. It is not known whether this ergodic
postulate can be derived from Hamilton’s equations.
Early computer simulations in the 1960s revealed that some non-linear
systems (with infinitely many degrees of freedom!) are not ergodic. Soliton
equations
ut = 6uux − uxxx ,
u = u(x, t),
KdV
or
φxx − φtt = sin φ,
φ = φ(x, t),
Sine-Gordon
are examples of such systems. Both possess infinitely many first integrals. We
shall study them in Chapter 2.
1.2
Integrability and action–angle variables
Given a system of Hamilton’s equations (1.1.2) it is often sufficient to know
n (rather than 2n − 1) first integrals as each of them reduces the order of the
system by two. This underlies the following definition of an integrable system.
Definition 1.2.1 An integrable system consists of a 2n-dimensional phasespace M together with n globally defined independent functions (in the sense
that the gradients ∇ f j are linearly independent vectors on the tangent space at
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1.2
Integrability and action–angle variables
any point in M) f1 , . . . , fn : M → R such that
{ f j , fk} = 0,
j, k = 1, . . . , n.
(1.2.3)
The vanishing of Poisson brackets (1.2.3) means that the first integrals are in
involution. We shall show that integrable systems lead to completely solvable
Hamilton’s equations of motion. Let us first explore the freedom in (1.1.2)
given by a coordinate transformation of phase space
Qk = Qk( p, q) and Pk = Pk( p, q).
This transformation is called canonical if it preserves the Poisson bracket
n
k=1
∂ f ∂g
∂ f ∂g
−
=
∂qk ∂ pk ∂ pk ∂qk
n
k=1
∂ f ∂g
∂ f ∂g
−
∂ Qk ∂ Pk ∂ Pk ∂ Qk
for all f, g : M −→ R. Canonical transformations preserve Hamilton’s
equations (1.1.2).
Given a function S(q, P, t) such that
det
∂2S
∂q j ∂ Pk
= 0,
we can construct a canonical transformation by setting
pk =
∂S
,
∂qk
Qk =
∂S
∂S
, and H = H +
.
∂ Pk
∂t
The function S is an example of a generating function [5, 102, 187]. The idea
behind the following theorem is to seek a canonical transformation such that
in the new variables H = H(P1 , . . . , Pn ) so that
Pk(t) = Pk(0) = const
and
Qk(t) = Qk(0) + t
∂H
.
∂ Pk
Finding the generating function for such canonical transformation is in practice
very difficult, and deciding whether a given Hamiltonian system is integrable
(without a priori knowledge of n Poisson commuting integrals) is still an open
problem.
Theorem 1.2.2 (Arnold–Liouville theorem [5]) Let
(M, f1 , . . . , fn )
be an integrable system with Hamiltonian H = f1 , and let
M f := {( p, q) ∈ M; fk( p, q) = ck},
ck = const,
be an n-dimensional level surface of first integrals fk. Then
k = 1, . . . , n
5
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6
1 : Integrability in classical mechanics
r If M f is compact and connected then it is diffeomorphic to a torus
T n := S 1 × S 1 × · · · × S 1 ,
and (in a neighbourhood of this torus in M) one can introduce ‘action–angle’
coordinates
I1 , . . . , In , φ1 , . . . , φn ,
0 ≤ φk ≤ 2π,
such that the angles φk are coordinates on M f and the actions Ik =
Ik( f1 , . . . , fn ) are first integrals.
r The canonical equations of motion (1.1.2) become
I˙k = 0 and φ˙ k = ωk(I1 , . . . , In ),
k = 1, . . . , n
(1.2.4)
and so the integrable systems are solvable by quadratures (a finite number of
algebraic operations and integrations of known functions).
Proof We shall follow the proof given in [5], but try to make it more accessible by avoiding the language of differential forms.
r The motion takes place on the surface
f1 ( p, q) = c1 , f2 ( p, q) = c2 , . . . , fn ( p, q) = cn
of dimension 2n − n = n. The first part of the theorem says that this surface
is a torus. 1 For each point in M there exists precisely one torus T n passing
through that point. This means that M admits a foliation by n-dimensional
leaves. Each leaf is a torus and different tori correspond to different choices
of the constants c1 , . . . , cn .
Assume
∂ fj
∂ pk
det
=0
so that the system fk( p, q) = ck can be solved for the momenta pi
pi = pi (q, c)
and the relations fi (q, p(q, c)) = ci hold identically. Differentiate these identities with respect to q j
∂ fi
+
∂q j
k
∂ fi ∂ pk
=0
∂ pk ∂q j
1 This part of the proof requires some knowledge of Lie groups and Lie algebras. It is given in
Appendix A.
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1.2
Integrability and action–angle variables
and multiply the resulting equations by ∂ fm/∂ p j
j
∂ fm ∂ fi
+
∂ p j ∂q j
∂ fm ∂ fi ∂ pk
= 0.
∂ p j ∂ pk ∂q j
j,k
Now swap the indices and subtract (mi) − (im). This yields
∂ fi ∂ fm ∂ pk
∂ fm ∂ fi ∂ pk
−
∂ p j ∂ pk ∂q j
∂ p j ∂ pk ∂q j
{ fi , fm} +
j,k
= 0.
The first term vanishes as the first integrals are in involution. Rearranging
the indices in the second term gives
∂ fi ∂ fm
∂ pk ∂ p j
j,k
∂ pk ∂ p j
−
∂q j
∂qk
=0
and, as the matrices ∂ fi /∂ pk are invertible,
∂ pk ∂ p j
−
= 0.
∂q j
∂qk
(1.2.5)
This condition implies that
p j dq j = 0
j
for any closed contractible curve on the torus T n . This is a consequence of
Stokes’ theorem. To see it recall that in n = 3
δD
p · dq =
(∇ × p) · dq
D
where δ D is the boundary of the surface D and
(∇ × p)m =
1
2
jkm
∂ pk ∂ p j
−
∂q j
∂qk
.
r There are n closed curves which cannot be contracted down to a point, so
that the corresponding integrals do not automatically vanish.
T
C1
n
C2
Cycles on a torus
7
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8
1 : Integrability in classical mechanics
Therefore we can define the action coordinates
Ik :=
1
2π
p j dq j ,
k
(1.2.6)
j
where the closed curve k is the kth basic cycle (the term ‘cycle’ in general
means ‘submanifold without boundary’) of the torus T n
k
= {(φ˜ 1 , . . . , φ˜ n ) ∈ T n ; 0 ≤ φ˜ k ≤ 2π, φ˜ j = const for j = k},
where φ˜ are some coordinates 2 on T n .
Stokes’ theorem implies that the actions (1.2.6) are independent of the choice
of k.
k
k
Stokes’ theorem
This is because
p j dq j +
k
j
ˆk
p j dq j =
j
∂ pj
∂ pi
−
∂q j
∂qi
dq j ∧ dqi = 0
where we have chosen and ˆ to have opposite orientations.
r The actions (1.2.6) are also first integrals as p(q, c)dq only depends on
ck = fk and the fk’s are first integrals. The actions are Poisson commuting
{Ii , I j } =
∂ Ii ∂ fr ∂ I j ∂ fs
∂ Ii ∂ fr ∂ I j ∂ fs
−
=
∂ fr ∂qk ∂ fs ∂ pk ∂ fr ∂ pk ∂ fs ∂qk
r,s,k
r,s
∂ Ii ∂ I j
{ fr , fs } = 0
∂ fr ∂ fs
and in particular {Ik, H} = 0.
The torus M f can be equivalently represented by
I1 = c˜1 , . . . , I1 = c˜n ,
2
This is a non-trivial step. In practice it is unclear how to explicitly describe the n-dimensional
torus and the curves k in 2n-dimensional phase space. Thus, to some extend the Arnold–Liouville
theorem has the character of an existence theorem.
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1.2
Integrability and action–angle variables
for some constants c˜1 , . . . , c˜n . (We might have been tempted just to define
Ik = fk, but then the transformation ( p, q) → (I, φ) would not be canonical
in general.)
r We shall construct the angle coordinates φk canonically to conjugate to the
actions using a generating function
q
p j dq j ,
S(q, I) =
q0
j
where q0 is some chosen point on the torus. This definition does not depend
on the path joining q0 and q as a consequence of (1.2.5) and Stokes’ theorem.
Choosing a different q0 just adds a constant to S thus leaving the angles
φi =
∂S
∂ Ii
invariant.
r The angles are periodic coordinates with a period 2π . To see this consider
two paths C and C ∪ Ck (where Ck represents the kth cycle) between q0 and
q and calculate
S(q, I) =
p j dq j =
C∪Ck
p j dq j +
C
j
p j dq j = S(q, I) + 2π Ik
Ck
j
j
so
φk =
∂S
= φk + 2π.
∂ Ik
T
C
n
q
0
q
Ck
Generating function
r The transformations
q = q(φ, I),
p = p(φ, I)
and
φ = φ(q, p),
I = I(q, p)
are canonical (as they are defined by a generating function) and invertible.
Thus,
{I j , Ik} = 0,
{φ j , φk} = 0, and {φ j , Ik} = δ jk
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10
1 : Integrability in classical mechanics
and the dynamics is given by
φ˙ k = {φk, H} and I˙k = {Ik, H},
where
H(φ, I) = H(q(φ, I), p(φ, I)).
The Ik’s are first integrals, therefore
∂H
=0
I˙k = −
∂φk
so H = H(I) and
∂H
= ωk(I)
φ˙ k =
∂ Ik
where the ωk’s are also first integrals. This proves (1.2.4). Integrating these
canonical equations of motion yields
φk(t) = ωk(I)t + φk(0)
and
Ik(t) = Ik(0).
(1.2.7)
These are n circular motions with constant angular velocities.
The trajectory (1.2.7) may be closed on the torus or it may cover it densely.
This depends on the values of the angular velocities. If n = 2 the trajectory will
be closed if ω1 /ω2 is rational and dense otherwise.
Interesting things happen to the tori under a small perturbation of the
integrable Hamiltonian
H(I) −→ H(I) + K(I, φ).
In some circumstances the motion is still periodic and most tori do not vanish
but become deformed. This is governed by the Kolmogorov–Arnold–Moser
theorem – not covered in this book. Consult the popular book by Schuster
[145], or read the complete account given by Arnold [5].
r Example. All time-independent Hamiltonian systems with two-dimensional
phase spaces are integrable. Consider the harmonic oscillator with the Hamiltonian
H( p, q) =
1 2
( p + ω2 q2 ).
2
Different choices of the energy E give a foliation of M by ellipses
1 2
( p + ω2 q2 ) = E.
2
