Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Malliavin Calculus A B Cruzeiro 383

Marsden–Weinstein Reduction see Cotangent Bundle Reduction: Poisson Reduction: Symmetry and

Symplectic Reduction

Maslov Index see Optical Caustics: Semiclassical Spectra and Closed Orbits: Stationary Phase

Approximation

Mathai–Quillen Formalism SWu 390

Mathematical Knot Theory L Boi 399

Matrix Product States see Finitely Correlated States

Mean Curvature Flow see Geometric Flows and the Penrose Inequality

Mean Field Spin Glasses and Neural Networks A Bovier 407

Measure on Loop Spaces H Airault 413

Metastable States S Shlosman 417

Minimal Submanifolds T H Colding and W P Minicozzi II 420

Minimax Principle in the Calculus of Variations A Abbondandolo 432

Mirror Symmetry: A Geometric Survey R P Thomas 439

Modular Tensor Categories see Braided and Modular Tensor Categories

Moduli Spaces: An Introduction F Kirwan 449

Multicomponent Fluids see Interfaces and Multicomponent Fluids

Multi-Hamiltonian Systems F Magri and M Pedroni 459

Multiscale Approaches A Lesne 465

Negative Refraction and Subdiffraction Imaging S O’Brien and S A Ramakrishna 483

Newtonian Fluids and Thermohydraulics G Labrosse and G Kasperski 492

Newtonian Limit of General Relativity J Ehlers 503

Noncommutative Geometry and the Standard Model T Schu

cker 509

Noncommutative Geometry from Strings Chong-Sun Chu 515

Noncommutative Tori, Yang–Mills, and String Theory A Konechny 524

Nonequilibrium Statistical Mechanics (Stationary): Overview G Gallavotti 530

Nonequilibrium Statistical Mechanics: Dynamical Systems Approach P Butta

and C Marchioro 540

Nonequilibrium Statistical Mechanics: Interaction between Theory and

Numerical Simulations R Livi 544

Nonlinear Schro

dinger Equations M J Ablowitz and B Prinari 552

Non-Newtonian Fluids C Guillope

560

Nonperturbative and Topological Aspects of Gauge Theory R W Jackiw 568

Normal Forms and Semiclassical Approximation D Bambusi 578

N-Particle Quantum Scattering D R Yafaev 585

Nuclear Magnetic Resonance P T Callaghan 592

Number Theory in Physics M Marcolli 600

Operads J Stasheff 609

Operator Product Expansion in Quantum Field Theory H Osborn 616

Optical Caustics A Joets 620

Optimal Cloning of Quantum States M Keyl 628

Optimal Transportation Y Brenier 632

Ordinary Special Functions W Van Assche 637

CONTENTS xlv

VOLUME 4

Painleve

Equations N Joshi 1

Partial Differential Equations: Some Examples R Temam 6

Path Integral Methods see Functional Integration in Quantum Physics; Feynman Path Integrals

Path Integrals in Noncommutative Geometry RLe

andre 8

Peakons D D Holm 12

Penrose Inequality see Geometric Flows and the Penrose Inequality

Percolation Theory V Beffara and V Sidoravicius 21

Perturbation Theory and Its Techniques R J Szabo 28

Perturbative Renormalization Theory and BRST K Fredenhagen and M Du

tsch 41

Phase Transition Dynamics A Onuki 47

Phase Transitions in Continuous Systems E Presutti 53

Pirogov–Sinai Theory R Kotecky

Point-Vortex Dynamics S Boatto and D Crowdy 66

Poisson Lie Groups see Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups

Poisson Reduction J-P Ortega and T S Ratiu 79

Polygonal Billiards S Tabachnikov 84

Positive Maps on C-Algebras F Cipriani 88

Pseudo-Riemannian Nilpotent Lie Groups P E Parker 94

q-Special Functions T H Koornwinder 105

Quantum 3-Manifold Invariants C Blanchet and V Turaev 117

Quantum Calogero–Moser Systems R Sasaki 123

Quantum Central-Limit Theorems A F Verbeure 130

Quantum Channels: Classical Capacity A S Holevo 142

Quantum Chromodynamics G Sterman 144

Quantum Cosmology M Bojowald 153

Quantum Dynamical Semigroups R Alicki 159

Quantum Dynamics in Loop Quantum Gravity H Sahlmann 165

Quantum Electrodynamics and Its Precision Tests S Laporta and E Remiddi 168

Quantum Entropy D Petz 177

Quantum Ergodicity and Mixing of Eigenfunctions S Zelditch 183

Quantum Error Correction and Fault Tolerance D Gottesman 196

Quantum Field Theory in Curved Spacetime B S Kay 202

Quantum Field Theory: A Brief Introduction L H Ryder 212

Quantum Fields with Indefinite Metric: Non-Trivial Models S Albeverio and H Gottschalk 216

Quantum Fields with Topological Defects M Blasone, G Vitiello and P Jizba 221

Quantum Geometry and Its Applications A Ashtekar and J Lewandowski 230

Quantum Group Differentials, Bundles and Gauge Theory T Brzezin

ski 236

Quantum Hall Effect K Hannabuss 244

Quantum Mechanical Scattering Theory D R Yafaev 251

Quantum Mechanics: Foundations R Penrose 260

Quantum Mechanics: Generalizations P Pearle and A Valentini 265

Quantum Mechanics: Weak Measurements L Dio

si 276

Quantum n-Body Problem R G Littlejohn 283

xlvi CONTENTS

Quantum Phase Transitions S Sachdev 289

Quantum Spin Systems B Nachtergaele 295

Quantum Statistical Mechanics: Overview L Triolo 302

Quasiperiodic Systems P Kramer 308

Quillen Determinant S Scott 315

Quivers see Finite-Dimensional Algebras and Quivers

Random Algebraic Geometry, Attractors and Flux Vacua M R Douglas 323

Random Dynamical Systems V Arau

jo 330

Random Matrix Theory in Physics T Guhr 338

Random Partitions A Okounkov 347

Random Walks in Random Environments L V Bogachev 353

Recursion Operators in Classical Mechanics F Magri and M Pedroni 371

Reflection Positivity and Phase Transitions Y Kondratiev and Y Kozitsky 376

Regularization for Dynamical -Functions V Baladi 386

Relativistic Wave Equations Including Higher Spin Fields R Illge and V Wu

nsch 391

Renormalization: General Theory J C Collins 399

Renormalization: Statistical Mechanics and Condensed Matter M Salmhofer 407

Resonances N Burq 415

Ricci Flow see Singularities of the Ricci Flow

Riemann Surfaces K Hulek 419

Riemann–Hilbert Methods in Integrable Systems D Shepelsky 429

Riemann–Hilbert Problem V P Kostov 436

Riemannian Holonomy Groups and Exceptional Holonomy D D Joyce 441

Saddle Point Problems M Schechter 447

Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools D Buchholz and

S J Summers 456

Scattering in Relativistic Quantum Field Theory: The Analytic Program J Bros 465

Scattering, Asymptotic Completeness and Bound States D Iagolnitzer and J Magnen 475

Schro

dinger Operators V Bach 487

Schwarz-Type Topological Quantum Field Theory R K Kaul, T R Govindarajan and P Ramadevi 494

Seiberg–Witten Theory Siye Wu 503

Semiclassical Approximation see Stationary Phase Approximation; Normal Forms and

Semiclassical Approximation

Semiclassical Spectra and Closed Orbits Y Colin de Verdie

re 512

Semilinear Wave Equations P D’Ancona 518

Separation of Variables for Differential Equations S Rauch-Wojciechowski and K Marciniak 526

Separatrix Splitting D Treschev 535

Several Complex Variables: Basic Geometric Theory A Huckleberry and T Peternell 540

Several Complex Variables: Compact Manifolds A Huckleberry and T Peternell 551

Shock Wave Refinement of the Friedman–Robertson–Walker Metric B Temple and J Smoller 559

Shock Waves see Symmetric Hyperbolic Systems and Shock Waves

Short-Range Spin Glasses: The Metastate Approach C M Newman and D L Stein 570

Sine-Gordon Equation S N M Ruijsenaars 576

Singularities of the Ricci Flow M Anderson 584

Singularity and Bifurcation Theory J-P Franc¸ oise and C Piquet 588

CONTENTS xlvii

Sobolev Spaces see Inequalities in Sobolev Spaces

Solitons and Kac–Moody Lie Algebras E Date 594

Solitons and Other Extended Field Configurations R S Ward 602

Source Coding in Quantum Information Theory N Datta and T C Dorlas 609

Spacetime Topology, Causal Structure and Singularities R Penrose 617

Special Lagrangian Submanifolds see Calibrated Geometry and Special Lagrangian Submanifolds

Spectral Sequences P Selick 623

Spectral Theory of Linear Operators M Schechter 633

Spin Foams A Perez 645

Spin Glasses F Guerra 655

Spinors and Spin Coefficients K P Tod 667

VOLUME 5

Stability of Flows S Friedlander 1

Stability of Matter J P Solovej 8

Stability of Minkowski Space S Klainerman 14

Stability Problems in Celestial Mechanics A Celletti 20

Stability Theory and KAM G Gentile 26

Standard Model of Particle Physics G Altarelli 32

Stationary Black Holes R Beig and P T Chrus

ciel 38

Stationary Phase Approximation J J Duistermaat 44

Statistical Mechanics and Combinatorial Problems R Zecchina 50

Statistical Mechanics of Interfaces S Miracle-Sole

Stochastic Differential Equations F Russo 63

Stochastic Hydrodynamics B Ferrario 71

Stochastic Loewner Evolutions G F Lawler 80

Stochastic Resonance S Herrmann and P Imkeller 86

Strange Attractors see Lyapunov Exponents and Strange Attractors

String Field Theory L Rastelli 94

String Theory: Phenomenology A M Uranga 103

String Topology: Homotopy and Geometric Perspectives R L Cohen 111

Superfluids D Einzel 115

Supergravity K S Stelle 122

Supermanifolds F A Rogers 128

Superstring Theories C Bachas and J Troost 133

Supersymmetric Particle Models S Pokorski 140

Supersymmetric Quantum Mechanics J-W van Holten 145

Supersymmetry Methods in Random Matrix Theory M R Zirnbauer 151

Symmetric Hyperbolic Systems and Shock Waves S Kichenassamy 160

Symmetries and Conservation Laws L H Ryder 166

Symmetries in Quantum Field Theory of Lower Spacetime Dimensions J Mund and K-H Rehren 172

Symmetries in Quantum Field Theory: Algebraic Aspects J E Roberts 179

Symmetry and Symmetry Breaking in Dynamical Systems I Melbourne 184

Symmetry and Symplectic Reduction J-P Ortega and T S Ratiu 190

Symmetry Breaking in Field Theory T W B Kibble 198

Symmetry Classes in Random Matrix Theory M R Zirnbauer

204

Synchronization of Chaos M A Aziz-Alaoui 213

xlviii CONTENTS

t Hooft–Polyakov Monopoles see Solitons and Other Extended Field Configurations

Thermal Quantum Field Theory CDJa

kel 227

Thermohydraulics see Newtonian Fluids and Thermohydraulics

Toda Lattices Y B Suris 235

Toeplitz Determinants and Statistical Mechanics E L Basor 244

Tomita–Takesaki Modular Theory S J Summers 251

Topological Defects and Their Homotopy Classification T W B Kibble 257

Topological Gravity, Two-Dimensional T Eguchi 264

Topological Knot Theory and Macroscopic Physics L Boi 271

Topological Quantum Field Theory: Overview J M F Labastida and C Lozano 278

Topological Sigma Models D Birmingham 290

Turbulence Theories R M S Rosa 295

Twistor Theory: Some Applications L Mason 303

Twistors K P Tod 311

Two-Dimensional Conformal Field Theory and Vertex Operator Algebras M R Gaberdiel 317

Two-Dimensional Ising Model B M McCoy 322

Two-Dimensional Models B Schroer 328

Universality and Renormalization M Lyubich 343

Variational Methods in Turbulence F H Busse 351

Variational Techniques for Ginzburg–Landau Energies S Serfaty 355

Variational Techniques for Microstructures G Dolzmann 363

Vertex Operator Algebras see Two-Dimensional Conformal Field Theory and Vertex

Operator Algebras

Viscous Incompressible Fluids: Mathematical Theory J G Heywood 369

von Neumann Algebras: Introduction, Modular Theory, and Classification Theory V S Sunder 379

von Neumann Algebras: Subfactor Theory Y Kawahigashi 385

Vortex Dynamics M Nitsche 390

Vortices see Abelian Higgs Vortices: Point-Vortex Dynamics

Wave Equations and Diffraction M E Taylor 401

Wavelets: Application to Turbulence M Farge and K Schneider 408

Wavelets: Applications M Yamada 420

Wavelets: Mathematical Theory K Schneider and M Farge 426

WDVV Equations and Frobenius Manifolds B Dubrovin 438

Weakly Coupled Oscillators E M Izhikevich and Y Kuramoto 448

Wheeler–De Witt Theory J Maharana 453

Wightman Axioms see Axiomatic Quantum Field Theory

Wulff Droplets S Shlosman 462

Yang–Baxter Equations J H H Perk and H Au-Yang 465

INDEX 475

CONTENTS xlix

PERMISSION ACKNOWLEDGMENTS

The following material is reproduced with kind permission of Nature Publishing Group

Figures 11 and 12 of ‘‘Point-vortex Dynamics’’

http://www.nature.com/nature

The following material is reproduced with kind permission of Oxford University Press

Figure 1 of ‘‘Random Walks in Random Environments’’

http://www.oup.co.uk

Introductory Articles

Introductory Article: Classical Mechanics

G Gallavotti, Universita

di Roma ‘‘La Sapienza,’’

Rome, Italy

ª 2006 G Gallavotti. Published by Elsevier Ltd.

General Principles

Classical mechanics is a theory of motions of point

particles. If X = (x

, ..., x

) are the particle positions

in a Cartesian inertial system of coordinates, the

equations of motion are determined by their masses

, ..., m

), m

> 0, and by the potential energy of

interaction, V(x

, ..., x

), as

€

¼@

Vðx

; ...; x

Þ; i ¼ 1; ...; n ½1

here x

= (x

, ..., x

) are coordinates of the ith

particle and @

is the gradient (@

, ..., @

); d is the

space dimension (i.e., d = 3, usually). The potential

energy function will be supposed ‘‘smooth,’’ that is,

analytic except, possibly, when two positions coin-

cide. The latter exception is necessary to include the

important cases of gravitational attraction or, when

dealing with electrically charged particles, of Cou-

lomb interaction. A basic result is that if V is

bounded below, eqn [1] admits, given initial data

= X(0),

X(0), a unique global solution

t !X(t), t 2 (1, 1); otherwise a solution can fail

to be global if and only if, in a finite time, it reaches

infinity or a singularity point (i.e., a configuration in

which two or more particles occupy the same point:

an event called a collision).

In eqn [1], @

V(x

, ..., x

) is the force acting on

the points. More general forces are often admitted.

For instance, velocity-dependent friction forces: they

are not consid ered here because of their phenomeno-

logical nature as models for microscopic phenomena

which should also, in principle, be explained in

terms of conservative forces (furthermore, even from

a macroscopic viewpoi nt, they are rather incomplete

models, as they should be considered together with

the important heat generation phenomena that

accompany them). Another interesting example of

forces not corresponding to a potential are certain

velocity-dependent forces like the Coriolis force

(which, however, appears only in noninertial frames

of reference) and the closely related Lorentz force

(in electromagnetism): they could be easily accom-

modated in the Hamiltonian formulation of

mechanics; see Appendix 2.

The action principle states that an equivalent

formulation of the eqns [1] is that a motion

t !X

(t) satisfying [1] during a time interval

, t

] and leading from X

= X

)toX

= X

renders stationary the action

AðfXgÞ ¼

i¼1

ðtÞ

 VðXðtÞÞ

dt ½2

within the class M

, t

, X

) of smooth (i.e.,

analytic) ‘‘motions’’ t !X(t) defined for t 2 [t

, t

]

and leading from X

to X

The function

LðY, XÞ¼

i¼1

 VðXÞ¼

def

KðYÞVðXÞ,

Y ¼ðy

, ..., y

is called the Lagrangian function and the action can

be written as

Lð

XðtÞ; XðtÞÞdt

The quantity K(

X(t)) is called kinetic energy and

motions satisf ying [1] conserve energy as time

t varies, that is,

Kð

XðtÞÞ þ VðXðtÞÞ ¼ E ¼ const : ½3

Hence the action principle can be intuitively thought

of as saying that motions proceed by keeping

constant the energy, sum of the kinetic and potential

energies, while trying to share a s evenly as possible

their (average over time) contribution to the energy.

In the special case in which V is translation invariant,

motions conserve linear momentum Q =

def

;ifV

is rotation invariant around the origin O, motions

conserve angular momentum M =

def

,where^

denotes the vector product in R

, that is, it is the tensor

(a ^ b)

= a

 b

, i, j = 1, ..., d: if the dimension

d = 3thea ^ b will be naturally regarded as a vector.

More generally, to any continuous symmetry group of

the Lagrangian correspond conserved quantities: this is

formalized in the Noether theorem.

It is convenient to think that the scalar product

in R

is defined in terms of the ordinary scalar product

in R

, a  b =

j = 1

,by(v, w) =

i = 1

 w

so that kinetic energy and line element ds can be

written as K(

X) =

(

X)andds

i = 1

respectively. Therefore, the metric generated by the

latter scalar product can be called kinetic energy

metric.

The interest of the kinetic metric appears from the

Maupertuis’ principle (equivalent to [1]): the princi-

ple allows us to identify the trajectory traced in R

by a motion that leads from X

to X

moving with

energy E. Parametrizing such trajectories as

 !X() by a parameter  varying in [0, 1] so that

the line element is ds

= (@



X, @



X)d

, the principle

states that the trajectory of a motion with energy E

which leads from X

to X

makes stationary, among

the analytic curves x 2M

0, 1

, X

), the function

LðxÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

E  VðxðsÞÞ

ds ½4

so that the possible trajectories traced by the

solutions of [1] in R

and with energy E can be

identified with the geodesics of the metric

def

(E V(X))  ds

For more details, the reader is referred to Landau

and Lifshitz (1976) and Gallavotti (1983) .

Constraints

Often particles are subject to constraints which force

the motion to take place on a surface M  R

,i.e.,

X(t) is forced to be a point on the manifold

M. A typical example is provided by rigid systems

in which motions are subject to forces which keep

the mutual distances of the particles constant:

 x

j= 

,with

time-independent positive quan-

tities. In essentially all cases, the forces that imply

constraints, called constraint reactions, are velocity

dependent and, therefore, are not in the class of

conservative forces considered here, cf. [1]. Hence,

from a fundamental viewpoint admitting only conser-

vative forces, constrained systems should be regarded

as idealizations of systems subject to conservative

forces which approximately imply the constraints.

In general, the ‘-dimensional manifold M will not

admit a global system of coordinates: however, it

will be possible to describe points in the vicinity

of any X

2M by using N = nd coordinates

q = (q

, ..., q

‘

, q

‘þ1

, ..., q

) varying in an open ball

: X = X(q

, ..., q

‘

, q

‘þ1

, ..., q

The q-coordinates can be chosen well adapted to

the surface M and to the kinetic metric, i.e., so that

the points of M are identified by q

‘þ1

= = q

= 0

(which is the meaning of ‘‘adapted’’); furthermore,

infinitesimal displacements (0, ...,0,d"

‘þ1

, ...,d"

)

out of a point X

2 M are orthogonal to M (in the

kinetic metric) and have a length independent of the

position of X

on M (which is the meaning of ‘‘well

adapted’’ to the kinetic metric).

Motions constrained on M arise when the

potential V has the form

VðXÞ¼V

ðXÞþWðXÞ½5

where W is a smooth function which reaches its

minimum value, say equal to 0, precisely on the

manifold M while V

is another smooth potential.

The factor >0 is a parameter called the rigidity of

the constraint.

A particularly interesting case arises when the level

surfaces of W also have the geometric property of

being ‘‘parallel’’ to the surface M: in the precise sense

that the matrix @

W(X), i, j >‘is positive definite

and X-independent, for all X 2M, in a system of

coordinates well adapted to the kinetic metric.

A potential W with the latter properties can be

called an approximately ideal constraint reaction. In

fact, it can be proved that, given an initial datum

2 M with velocity

tangent to M, i.e., given

an initial datum whose coordinates in a local system

of coordinates are (q

, 0) and (

, 0) with q

, ..., q

0‘

) and

= (

, ...,

0‘

), the motion

generated by [1] with V given by [5] is a motion

t !X



(t) which

1. as  !1tends to a motion t !X

(t);

2. as long as X

(t) stays in the vicinity of the initial

data, say for 0  t  t

, so that it can be

described in the above local adapted coordinates,

its coordinates have the form t !(q(t), 0) =

(t), ..., q

‘

(t), 0, ..., 0): that is, it is a motion

developing on the constraint surface M; and

3. the curve t !X

(t), t 2 [0, t

], as an element of

the space M

0, t

, X

)) of analytic curves on

M connecting X

to X

), renders the action

AðXÞ¼

Kð

XðtÞÞ  V

ðXðtÞÞ



dt ½6

stationary.

2 Introductory Article: Classical Mechanics

The latter property can be formulated ‘‘intrinsically,’’

that is, referring only to M as a surface, via the

restriction of the metric ds

to line elements ds =

(dq

, ...,dq

‘

,0,..., 0) tangent to M at the point

X = (q

,0,...,0)2 M;wewriteds

1,‘

i, j

(q)

.The‘  ‘ symmetric positive-definite matrix g

canbecalledthemetric on M induced by the kinetic

energy. Then the action in [6] can be written as

AðqÞ¼

1;‘

i;j

ðqðtÞÞ

ðtÞ



ðqðtÞÞ

dt ½7

where

(q) =

def

(X(q

, ..., q

‘

,0, ..., 0)): the function

Lðh; qÞ¼

def

1;‘

i;j

ðqÞ



 V

ðqÞ



gðqÞh  h 

ðqÞ½8

is called the constrained Lagrangian of the system.

An important property is that the constrained motions

conserve the energy defined as E =

(g(q)

q)þ

(q); see next section.

The constrained motion X

(t)ofenergyE satisfies

the Maupertuis’ principle in the sense that the curve

on M on which the motion develops renders

LðxÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

E  V

ðx ðsÞÞ

ds ½9

stationary among the (smooth) curves that develop

on M connecting two fixed values X

and X

. In the

particular case in which ‘ = n this is again Mauper-

tuis’ principle for unconstrained motions under the

potential V(X). In general, ‘ is called the number of

degrees of freedom because a complete description

of the initial data requires 2‘ coordinates q(0),

q(0).

If W is minimal on M but the condition on W of

having level surfaces parallel to M is not satisfied, i.e.,

if W is not an approximate ideal constraint reaction,

it still remains true that the limit motion X

(t) takes

place on M. However, in general, it will not satisfy the

above variational principles. For this reason, motions

arising as limits (as  !1) of motions developing

under the potential [5] with W having minimum on M

and level curves parallel (in the above sense) to M are

called ideally constrained motions or motions subject

by ideal constraints to the surface M.

As an example, suppose that W has the form

W(X) =

i, j2P

(jx

 x

j)withw

(jxj)  0anana-

lytic function vanishing only when jxj= 

for i, j in

somesetofpairsP and for some given distances 

(e.g.,

(x) = (x

 

)

, >0). Then W can be shown to

satisfy the mentioned conditions and therefore, the so

constrained motions X

(t) of the body satisfy the

variational principles mentioned in connection with [7]

and [9]: in other words, the above natural way of

realizing a rather general rigidity constraint is ideal.

The modern viewpoint on the physical meaning of

the constraint reactions is as follows: looking at

motions in an inertial Cartesian system, it will appear

that the system is subject to the applied forces with

potential V

(X) and to constraint forces which are

defined as the differences R

= m

€

þ ¶

(X). The

latter reflect the action of the forces with potential

W(X) in the limit of infinite rigidity ( !1).

In applications, sometimes the action of a constraint

can be regarded as ideal: the motion will then verify the

variational principles mentioned and R can be com-

puted as the differences between the m

€

and the active

forces ¶

(X). In dynamics problems it is, however,

a very difficult and important matter, particularly in

engineering, to judge whether a system of particles can

be considered as subject to ideal constraints: this leads

to important decisions in the construction of machines.

It simplifies the calculations of the reactions and fatigue

of the materials but a misjudgment can have serious

consequences about stability and safety. For statics

problems, the difficulty is of lower order: usually

assuming that the constraint reaction is ideal leads to

an overestimate of the requirements for stability of

equilibria. Hence, employing the action principle to

statics problems, where it constitutes the principle of

virtual work, generally leads to economic problems

rather than to safety issues. Its discovery even predates

Newtonian mechanics.

We refer the reader to Arnol’d (1989) and

Gallavotti (1983) for more details.

Lagrange and Hamilton Forms

of the Equations of Motion

The stat ionarity condition for the action A(q), cf.

[7], [8], is formulated in terms of the Lagrangian

L(h, x), see [8],by



Lð

qðtÞ; qðtÞÞ

¼ @

Lð

qðtÞ; qðtÞÞ; i ¼ 1; ...;‘ ½10

which is a second-order differential equation called

the Lagrangian equation of motion. It can be cast in

‘‘normal form’’: for this purpose, adopting the

convention of ‘‘summation over repeated indices,’’

introduce the ‘‘generalized momenta’’

def

gðqÞ

; i ¼ 1; ...;‘ ½11

Introductory Article: Classical Mechanics 3

Since g(q) > 0, the motions t !q(t) and the corre-

sponding velocities t !

q(t) can be described equiva-

lently by t !(q(t), p(t)): and the equations of motion

[10] become the first-order equations

¼ @

Hðp; qÞ;

¼@

Hðp; qÞ½12

where the function H, called the Hamiltonian of the

system, is defined by

Hðp; qÞ¼

def

ðgðqÞ

1

p; pÞþV

ðqÞ½13

Equations [12], regarded as equations of motion for

phase space points ( p, q), are called Hamilton

equations. In general, q are local coordinates on M

and motions are specified by giving q,

q or p, q.

Looking for a coordinate-free representation of

motions consider the pairs X, Y with X 2 M and Y a

vector Y 2 T

tangent to M at the point X. The

collection of pairs (Y, X) is denoted T(M) = [

X2M

 {X}) and a motion t !(

X(t), X(t)) 2 T(M)in

local coordinates is represented by (

q(t), q(t)). The

space T(M) can be called the space of initial data for

Lagrange’s equations of motion: it has 2‘ dimen-

sions (also known as the ‘‘tangent bundle’’ of M).

Likewise, the space of initial data for the

Hamilton equations will be denoted T



(M) and it

consists of pairs X, P with X 2 M and P = g(X)Y

with Y a vector tangent to M at X. The space T



(M)

is called the phase space of the system: it has

2‘ dimensions (and it is occasionally called the

‘‘cotangent bundle’’ of M).

Immediate consequence of [12] is

HðpðtÞ; qðtÞÞ  0

and it means that H(p(t), q(t)) is constant along

the solutions of [12].NotingthatH(p, q) =

(1=2)(g(q)

q) þ

(q) is the sum of the kinetic

and potential energies, it follows that the conservation

of H along solutions means energy conservation in

presence of ideal constraints.

Let S

be the flow generated on the phase space

variables (p, q) by the solutions of the equations of

motion [12], that is, let t !S

(p, q)  (p(t), q(t))

denote a solution of [12] with initial data (p, q).

Then a (measurable) set  in phase space evolves in

time t into a new set S

 with the same volume: this

is obvious because the Hamilton equations [12] have

manifestly zero divergence (‘‘Liouville’s theorem’’).

The Hamilton equations also satisfy a variational

principle, called the Hamilton action principle: that

is, if M

, t

((p

, q

), (p

, q

); M) denotes the space of

the analytic functions j : t !(p(t), k(t)) which in the

time interval [t

, t

] lead from (p

, q

)to(p

, q

then the condition that j

(t) = (p(t), q(t)) satisfies

[12] can be equivalently formulated by requiring

that the function

ðjÞ¼

def



pðtÞ

kðtÞHðpðtÞ; kðtÞÞ



dt ½14

be stationary for j = j

:infact,eqns [12] are the

stationarity conditions for the Hamilton action

[14] on M

, t

((p

, q

), (p

, q

); M). And, since the

derivatives of p(t) do not appear in [14],statio-

narity is even achieved in the larger space

, t

, q

; M) of the motions j : t !(p(t), k(t))

leading from q

to q

without any r estriction on

the initial and final momenta p

, p

(which, there-

fore, cannot be pre scribed aprioriindependently

of q

, q

). If the prescribed data p

, q

, p

, q

are

not compatible with the equations of motion (e.g.,

H(p

, q

) 6¼ H(p

, q

)), then the action functional

has no stationary trajectory in M

, t

((p

, q

); M).

For more details, the reader is referred to Landau

and Lifshitz (1976), Arnol’d (1989),andGallavotti

(1983).

Canonical Transformations of Phase

Space Coordinates

The Hamiltonian form, [13], of the equations of

motion turns out to be quite useful in several

problems. It is, therefore, important to remark that

it is invariant under a special class of transformations

of coordinates, called canonical transformations.

Consider a local change of coordinates on phase

space, i.e., a smooth, smoothly invertible map

C(p, k) = (p

, k

) between an open set U in the

phase space of a Hamiltonian system with

‘ degrees of freedom, into an open set U

in a

2‘-dimensional space. The change of coordinates is

said to be canonical if for any solution

t !(p(t), k(t)) of equations like [12], for any

Hamiltonian H(p, k) defined on U, the C–image

t !(p

(t), k

(t)) = C(p(t), k (t)) is a solution of [12]

with the ‘‘same’’ Hamiltonian, that is, with

Hamiltonian H

, k

) =

def

H(C

1

, k

)).

The condition that a transformation of coordi-

nates is canonical is obtained by using the

arbitrariness of the function H and is simply

expressed as a necessary and sufficient property of

the Jacobian L,

L ¼



¼ @



; B

¼ @





;

¼ @





; D

¼ @



½15

4 Introductory Article: Classical Mechanics