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which can be shown to be smooth on I

. The

physical interpretation of this tensor field is based

on the following properties. In source-free regi ons

the field satisfies the spin-2 zero-rest-mass equation

bcd

¼ 0

which is very similar to the Maxwell equations for

the electromagnetic (spin-1) Faraday tensor. Thus,

bcd

is interpreted as the gravitational field, which

describes the gravitational waves contained inside

the system. The zero-rest-mass equation for K

bcd

and the fact that the field is smooth on I implies that

the Weyl tensor satisfies the ‘‘peeling’’ property. This

is a characteristic conspiracy between the fall-off

behavior of certain components of the Weyl tensor

along outgoing g-null-geodesics approaching I

M with respect to an affine parameter s for s !1

and their algebraic type. Symbolically, the Weyl

tensor has the following behavior as s !1 along

the null geodesic:

C ¼

½4

½31

½211

½1111

þ Oðs

5

Þ½5

where the numerator of each component indi cates

its Petrov type. The repeated principal null direction

(PND) in the first three components and one of the

PNDs in the fourth component are aligned with the

tangent vector of the geodesic. This implies that

the farthest reaching component of the Weyl tensor,

which is O(1=s), has the Petrov type of a radiati on

field. It is customary to combine the components

which are O(1=s

) into one complex function and

denote it by

5i

. When expressed in terms of the

field K

bcd

M, this fall-off behavior implies that

of all components of K

bcd

only

does not

necessarily vanish on I

In special cases like the Minkow ski, Schwarzs-

child, Kerr, and more generally in all asymptotically

flat stationary spacetimes, even

vanishes on I

For these reasons,

is called the radiation field of

the system, that is, that part of the gravitational field

which can be registered by the observers at infinity.

It describes the outgoing radiation which is being

emitted by the system during its evolution.

The Bondi–Sachs Mass-Loss Formula

Gravitational waves carry away energy from the

system. This is a consequence of the Bondi–Sachs

mass-loss formula. The Bondi–Sachs energy–

momentum is related to a weighted integral over a

cut C,

½W¼

4G

þ 

½d

S ½6

The quantity in brackets, the mass aspect, is a

combination of the scalar

which in a sense

measures the strength of the Coulomb-like part of

the gravitational field on I

and the complex

quantity . In a so-called Bondi coordinate system,

this quantity is related to the radiation field

the relation

¼

€



the dot indicating differentiation with respect to the

affine parameter along the null generators. Thus, 

is essentially the second time integral of the

radiation field. The mass aspect is integrated against

a function W which is an asymptotic translation,

that is, a linear combination of the first four

spherical harmonics. Thus, one can view the

expression [6] as defin ing a linear map T ! R.

Since T and M are isometric this defines a covector

on M, which can always be shown to be timelike,

 0. This positivity property together with the

fact that in the special cases of Schwarzschild and

Kerr spacetimes the integral yields the mass para-

meters when evaluated for a time translation

(W = 1) motivates the interpretation of P

as the

energy–momentum 4-vector of the spacetime at the

instant defined by the cut C. In particular, for W = 1

the integral gives the time component of P

, the

Bondi–Sachs energy E.

The interpretation of [6] as energy–momentum is

strengthened by the fact that P

arises as dual to the

translations which is familiar from Lagrangian field

theories where energy and momentum appear as

generators for time and space translations. In fact,

one can set up a Hamiltonian framework where the

role of the Bondi–Sachs energy–momentum as

generator of asymptotic translations is made

explicit.

This point of view suggests that one should also

be able to define a notion of angular momentum for

asymptotically flat spacetimes because angular

momentum arises as the generator of rotations,

which can also be defined asymptoti cally. However,

while there is a unique notion of translation on I

this is not the case for rotations (and boosts). The

reason is hidden in the structure of the BMS group

where the Lorentz group appears naturally as a

factor group but not as a unique subgroup. In

physical terms, the angular momentum depends on

an origin but there is no natural way to choose an

origin on I

. This ambiguity in the choice of origin

leads to sever al noneq uivalent expressions for

angular momentum in the literature.

Consider now two cuts C and C

, with C

later than

C. Then we may compute the difference E = E  E

of the Bondi–Sachs energies with respect to the two

Asymptotic Structure and Conformal Infinity 225

cuts. It turns out that this difference can be

expressed as an integral over the (three-dimensional)

piece  of I

which is bounded by the two cuts

(i.e., @=C

C):

 E ¼

4G



_

 d

V ½7

This result means that the Bondi–Sachs energy of the

system decreases, since E

< E and the rate of

decrease is given by the (positive-definite) amount

of gravitational radiation which leav es the system

during the period defined by the two cuts.

It is necessary to point out that in this article the

structure of null infinity has been postulated based

on physical reasonings. The Einstein equations have

been used only in a very weak sense, namely only in

a neighborhood of I . It is an entirely different

question whether the field equations are compatible

with this postulated structure. To answer it, one

needs to show that there are global solutions of the

Einstein equations which exhibit the postulated

behavior in the asymptotic region. This question

has been settled recently in the affirmative: there are

many global spacetimes which are asymptotically

flat in the sense described here.

This article discussed has the notion of null

infinity, that is, of spacetimes which are asymptoti-

cally flat in lightlike directions. Spacetimes which

are asymptotically flat in spacelike directions have

not been covered. The latter is a notion which has

been developed largely independently of null infinity

since it is essentially a property of an initial data set

and not of the entire four-dimensional spacetime.

Ultimately, these two notions should coincide, in the

sense that if one has an initial data set which is

asymptotically flat in spatial directions in an appro-

priate sense then its Cauchy development will be an

asymptotically flat spacetime. However, as of yet, it

is not clear what the appropriate conditions should

be because the structure of the gravitational field in

the neighborhood of spacelike infinity i

is not

sufficiently well understood so far.

See also: Black Hole Mechanics; Boundaries for

Spacetimes; Canonical General Relativity; Einstein

Equations: Exact Solutions; Einstein Equations: Initial

Value Formulation; General Relativity: Overview;

Gravitational Waves; Quantum Entropy; Spacetime

Topology, Causal Structure and Singularities; Stability of

Minkowski Space; Stationary Black Holes.

Further Reading

Ashtekar A (1987) Asymptotic Quantization. Naples: Bibliopolis.

Bondi H, van der Burg MGJ, and Metzner AWK (1962)

Gravitational waves in general relativity VII. Waves from

axi-symmetric isolated systems. Proceedings of the Royal

Society of London, Series A 269: 21–52.

Frauendiener J (2004) Conformal infinity. Living Reviews in

Relativity, vol. 3. http://relativity.livingreviews.org/Articles/

lrr-2004-1/index.html.

Friedrich H (1992) Asymptotic structure of space-time. In: Janis AI

and Porter JR (eds.) Recent Advances in General Relativity.

Boston: Birkha¨user.

Friedrich H (1998a) Einstein’s equation and conformal structure.

In: Huggett SA, Mason LJ, Tod KP, Tsou SS, and Woodhouse

NMJ (eds.) The Geometric Universe: Science, Geometry and

the Work of Roger Penrose. Oxford: Oxford University Press.

Friedrich H (1998b) Gravitational fields near space-like and null

infinity. Journal of Geometry and Physics 24: 83–163.

Geroch R (1977) Asymptotic structure of space-time. In: Esposito

FP and Witten L (eds.) Asymptotic Structure of Space-Time.

New York: Plenum.

Hawking S and Ellis GFR (1973) The Large Scale Structure of

Space-Time. Cambridge: Cambridge University Press.

Penrose R (1965) Zero rest-mass fields including gravitation:

asymptotic behaviour. Proceedings of the Royal Society of

London, Series A 284: 159–203.

Penrose R (1968 ) Structure of space-time. In: DeWitt CM

and Whe eler JA (eds.) Battel le Rencontres.NewYork:

W. A. Benjamin.

Penrose R and Rindler W (1984, 1986) Spinors and Space-Time,

Cambridge: Cambridge University Press.

Sachs RK (1962) Gravitational waves in general relativity VIII.

Waves in asymptotically flat space-time. Proceedings of the

Royal Society of London, Series A 270: 103–127.

Averaging Methods

A I Neishtadt, Russian Academy of Sciences,

Moscow, Russia

Introduction

Averaging methods are the methods of perturbation

theory that are based on the averaging principle and

the idea of dividing the dynamics into slow drift and

fast oscillations. The most common field of applica-

tions of averaging methods is the analysis of the

behavior of dynamical systems that differ from

integrable systems by small perturbations.

Averaging Principle

Equations of motion of a system that differ from an

integrable system by small perturbations often can

be written in the form

226 Averaging Methods

I ¼ "gðI;’;"Þ;

’ ¼ !ðIÞþ"f ðI;’;"Þ

I ¼ðI

; ...; I

Þ2R

’ ¼ð’

; ...;’

Þ2T

modd 2; 0 <" 1

½1

The small parameter " characterizes the amplitude

of the perturbation. For " = 0 one gets the

unperturbed system. The equation I = const. sin-

gles out an invariant m-dimensional torus of the

unperturbed system. The motion on this torus is

quasiperiodic with frequency vector !(I); compo-

nents of vector I are called ‘‘slow variables’’

whereas components of vector ’ are called ‘‘fast

variables’’ or ‘‘phases.’’ The rig ht-hand sides of

system [1] are 2-periodic with respect to all ’

.It

is assumed that they are smooth enough functions

of all arguments. It is also assumed that compo-

nents of the frequency v ector are not linearly

dependent over the ring of i nteger numbers

identically with respect to I. System [1] is called

a ‘‘system with rotating phases.’’

In applications, one is often interested mainly in

the behavior of slow variables. The ‘‘averaging

principle’’ (or method) consists in replacing the

system of perturbed equations [1] by the ‘‘averaged

system’’

J ¼ "GðJÞ; GðJÞ¼ð2Þ

m

gðJ;’;0Þd’ ½2

for the purpose of providing an approximate

description of the evolution of the slow variables

over time intervals of order 1=" or longer. Here,

d’ = d’

d’

. System [2] contains only slow

variables and, therefore, is much simpler for

investigation than system [1]. When passing from

system [1] to system [2], one ignores the terms

g(I, ’,0) G(I) on the right-hand side of [1]. The

averaging principle is based on the idea that these

terms oscillate and lead only to small oscillations

which are superimposed on the drift described by

the averaged system. To justify the averaging

principle, one should establish a relation between

the behavior of the solutions of systems [1] and [2].

This problem is still far from being completely

solved.

Another version of the averaging principles is

used in the case when frequencies are approxi-

mately in resonance. This means that one or

several relations of the form (k, !) = 0 approxi-

mately are valid with irreducible integer coefficient

vectors k 6¼ 0; here, (k, !) is the standard scalar

product in R

.Let be a sublattice of the integer

lattice Z

generated by these vectors. Let

r = rank  and k

(1)

, k

(2)

, ..., k

(m)

be a basis in Z

the f irst r vectors of which belong to . Instead of

’, one can introduce new variables:

# ¼ð#

; ...;#

Þ2T

modd 2

 ¼ð

; ...;

mr

Þ2T

mr

modd 2

¼ðk

ðiÞ

;’Þ;

¼ðk

ðrþjÞ

;’Þ

Let R be an r  m matrix whose rows are vectors

(i)

,1 i  r. For an approximate description of the

behavior of variables I, #, the averaging principle

prescribes replacing syst em [1] by the system

J ¼ "G



ðJ;Þ;

 ¼ R!ðJÞþ"RF



ðJ;Þ



ðJ;#Þ¼ð2Þ

ðmrÞ

mr

gðJ;’;0Þd



ðJ;#Þ¼ð2Þ

ðmrÞ

mr

f ðJ;’;0Þd

½3

(one should express g, f through #,  and then

integrate over ,d = d

d

mr

). System [3] is

called ‘‘partially averaged system’’ for resonances in

. Functions G



, F



can be obtained from Fourier

series expansions of functions g, f for " = 0

by throwing away harmonics exp (i(k, ’)), k =2

(nonresonant harmonics). Passing from system [1]

to system [3] is based on the idea that the ignored

nonresonant harmonics oscillate fast and do not

affect essentially the evolution of the slow variables.

Now let system [1] be a Hamiltonian system close

to an integrable one. The Hamiltonian function has

the form

H ¼ H

ðpÞþ"H

ðp;’;y; x;"Þ

where ’, x are coordinates and p, y are conjugated

to them. The equations of motion have the same

form as [1], with I = (p, y, x):

p ¼"

@’

;

y ¼"

x ¼ "

;

’ ¼

þ "

½4

The averaging principle in the case when there are

no resonant relations leads to the system

p ¼ 0;

y ¼"

;

x ¼ "

¼ð2Þ

m

ðp;’;y; x; 0Þd’

½5

Therefore, in this case there is no drift in p, and the

behavior of y, x is described by the Hamiltonian

system, which contains p as a parameter. Equations

of motion of planets around the Sun can be reduced

to the form [4]. The issue of the absence of the

evolution of momenta p is known in this problem as

Averaging Methods 227

the Lagrange–Laplace theorem, about the absence of

the evolution of semimajor axes of planetary orbits.

Elimination of Fast Variables, Decoupling

of Slow and Fast Motions

The basic role in the averaging method is played by

the idea that the exact system can be in the principal

approximation transformed into the averaged sys-

tem by means of a transformation of variables close

to the identical one. The extension of this idea is the

idea that similar transformation of variables allows

one to eliminate, up to an arbitrary degree of

accuracy, the fast phases from the right-hand sides

of the equations of perturbed motion and in this

way decouple the slow motion from the fast one.

For system [1], provided there are no resonant

relations between frequencies, the elimination of fast

variables is performed as follow s. The desirable

transformation of variables (I, ’) 7!(J, ) is sought

as a formal series

I ¼ J þ "u

ðJ; Þþ"

ðJ; Þþ

’ ¼ þ "v

ðJ; Þþ"

ðJ; Þþ

½6

where functions u

, v

are 2-periodic in . The

transformation [6] should be chosen in such a way

that in the new variables the right-hand sides of

equations of motion do not contain fast variables,

that is, the equations of motion should have the

form

J ¼ "G

ðJÞþ"

ðJÞþ

¼ !ðJÞþ"F

ðJÞþ"

ðJÞþ

½7

Substituting [6] into [7], taking into account [1], and

equating the terms of the same order in ", we obtain

the following set of relations:

ðJÞ¼gðJ; ;0Þ

ðJÞ¼f ðJ; ;0Þþ



ðJÞ¼X

ðJ; Þ

iþ1

ðJÞ¼Y

ðJ; Þþ

iþ1



iþ1

!; i  1

½8

The functions X

, Y

are uniquely determined by the

terms u

, v

, ..., u

, v

in expansion [6]. The first

equation in [8] implies that

ðJÞ¼g

ðJÞ¼GðJÞ

ðJ; Þ¼

k6¼0

iðk;!Þ

expðiðk; ÞÞ þ u

ðJÞ

½9

where g

, k 2 Z

, are Fourier coefficients of func-

tion g at " = 0, and u

is an arbitrary function of J.It

is assumed that the denominators in [9] do not

vanish, and that the series in [9] converges and

determines a smooth function. In the same way,

from the other equations in [8] one can sequentially

determine F

, v

, ..., G

, u

iþ1

, F

, v

iþ1

, i  1.

On truncating the series in [6] and [7] at the terms

of order "

, we obtain a truncated system of the lth

approximation. The equation for J is decoupled

from the other equations an d can be solved

separately. Then the behavior of is determined

by means of quadrature. The behavior of original

variable I in this approximation is a slow drift

(described by the equation for J), on which small

oscillations (described by transformation of variables)

are superimposed. The behavior of ’ can be repre-

sented as a rotation with slowly varying frequency,

on which oscillations are also superimposed. For l = 1,

the truncated system coincides with the averaged

system [2].

If the sublattice   Z

specifying possible

resonant relations is given, then in an analogous

manner one can construct a formal transformation

of variables (I, ’) 7!(J, ) such that, in the new

variables, the fast phase will appear on the right-

hand sides of the differential equations for the new

variables only in combinations (k, ), with k 2 

(see, e.g., Arnol’d et al. (1988)). Again, on truncat-

ing the series on the right-hand sides of the

differential equations for the new variables at the

terms of order "

, we obtain a truncated system of

the lth approximation. At l = 1, this truncated

system coincides with the partially averaged system

[3] (for some special choice of arbitrary functions

that are contained in the formulas for transformation

of variables). If the original system is a Hamiltonian

system of the form [4], then the transformation of

variables eliminating the fast phases from the right-

hand sides of the differential equations can be

chosen to be symplectic. The corresponding

procedures are called ‘‘Lindstedt method’’ and

‘ ‘Newcomb method’ ’ (nonresonant case for n = m),

‘‘Delaunay method’’ (resonant case for n = m), and

‘‘von Zeipel method’’ (resonant case for n  m)(see

Poincare´ (1957) and Arnol’d et al. (1988)).

The calculation of high-order terms in the

procedures of elimination of fast variables is rather

cumbersome. There are versions of these procedures

which are convenient for symbolic processors

(especially for Hamiltonian systems, e.g., the

Deprit–Hori method; Giacaglia 1972).

The averaging method consists in using the

averaged system for the description of motion in

the first approximation and the truncated systems

228 Averaging Methods

obtained by means of the procedures of elimination

of fast variables in the higher approximations,

together with the corresponding transformations of

variables.

Justification of the Averaging Method

To justify the averaging method, one should estab-

lish conditions under which the deviation of the

slow variables along the solutions of the exact

system from the solutions of the averaged system

with appropriate initial data on time intervals of

order 1=" or longer tends to 0 as " ! 0. It is

desirable to have estimates from the above for these

deviations. The estimates of deviations of the

solutions of the exact syst em from the solutions of

the truncated systems obtained by means of the

procedure of elimination of fast phases are impor-

tant as well. It can happen that there are ‘‘bad’’

initial data for which the slow component of the

solution of the exact system deviates from the

solution of the averaged system by a value of order

1 over time of order 1=". In this case, one should

have estimates from above for the measure of the set

of such ‘‘bad’’ initial data; on the complementary set

of initial data, one should have estimates from

above for the deviation of slow variables along the

solutions of the exact system from the solution of

the averaged system. These problems are currently

far from being completely solved. Some general

results are described in the follow ing.

Let functions !, f , g on the right-hand side of

system [1] be defined and bounded together with a

sufficient number of derivatives in the domain D{I} 

{’}  [0, "

]. Let J(t) be the solution of the

averaged system [2] with initial condition I

2 D.

Let (I(t), ’(t)) be the solution of the exact system [1]

with initial conditions (I

, ’

). So, I(0) = J(0). It is

assumed that the solution J(t) is defined and stays at

a positive distance from the boundary of D on the

time interval 0  t  K=" , K = const > 0.

If system [1] is a one-frequency system (m = 1),

and the frequency ! does not vanish in D, then for

0  t  K=" the solution (I(t ), ’(t)) is well defined,

and jI(t)  J(t)j < C ", C = const. > 0. For ! = 1, this

assertion was proved by P Fatou (1928) and, by a

different method, by L I Mandel’shtam and L D

Papaleksi (1934). This was historically the

first result on the justification of the averaging

method (Mintropol’skii 1971). There is a proof

based on the elimination of fast variables (see , e.g.,

Arnol’d (1983)). For a one-frequency system, higher

approximations of the procedure of elimination of

fast variables allow the description of the dynamics

with an accuracy of the order of any power in " on

time intervals of order 1=" (Bogolyubov and

Mitropol’skii 1961).

If system [1] is a multifrequency system (m  2), but

the vector of frequencies is constant and nonresonant,

then for any >

0 and small enough "<"

()itholds

that jI(t)  J(t)j < for 0  t  K=" (Bogolyubov

1945, Bogolyubov and Mitropol’skii 1961). If, in

addition, the frequencies satisfy the Diophantine

condition j(k, !)j > const jkj



for all k 2 Z

n{0}

and some >0, then one can choose  = O("). In

this case, higher approximations of the procedure of

elimination of fast variables allow one to describe

the dynamics with an accuracy of the order of any

power in " on time intervals of order 1=" (see, e.g.,

Arnol’d et al. (1988)).

If the system is a multifrequency system, and

frequencies are not constant (but depend on the slow

variables I), then due to the evolution of slow

variables the frequencies themselves are evolving

slowly. At certain time moments, they can satisfy

certain resonant relations. One of the phenomena

that can take place here is a capture into a

resonance; this capture leads to a large deviation of

the solutions of the exact and averaged systems.

However, the general Anosov averag ing theorem

(Anosov 1960) implies that if the frequencies ! are

nonresonant for almost all I, then for any >0, the

inequality jI(t)  J( t)j <is satisfied for 0  t  K="

for all initial data outside a set E(, ") whose

measure tends to 0 as " ! 0. In many cases, it

turns out that mes E(, ") = O(

ﬃﬃﬃ

=) (in particular,

the sufficient condition for the last estimate is that

rank(@!=@I) = m)(Arnol ’d et al. (1988)).

The knowledge about averaging in two-

frequency systems (m = 2) on time intervals, of order

of 1="

, is relatively more complete (see Arnol’d

(1983), Arnol’d et al. (1988),andLochak and

Meunier (1988)). For Hamiltonian and reversible

systems, the justification of the averaging method is

a by-product of Kolmogorov–Arnold–Moser (KAM)

theory. The KAM theory provides estimates of the

difference between the solutions of the exact and

averaged systems for majority of initial data on

infinite time interval 1 < t < þ1. For remaining

data this difference can grow because of Arnol’d

diffusion, but, in general, very slowly. According to

the Nekhoroshev theorem, this difference is small on

time intervals whose length grows exponentially when

the perturbation decays linearly (for an analytic

Hamiltonian if the unperturbed Hamiltonian is a

generic function, the so-called steep function).

Another aspect of justification of the averaging

method is establishing relations between invariant

manifolds of the exact and averaged systems.

Consider, in particular, the case of a one-frequency

Averaging Methods 229

system and a multifrequency system with constant

Diophantine frequencies. Suppose that the averaged

system has an equilibrium such that real parts of all

its eigenvalues are different from 0, or a limit cycle

such that the absolute values of all but one of its

multipliers are different from 1. Then the exact

system has an invariant torus, respectively, m-or

(m þ 1)-dimensional, whose projection onto the

space of the slow variables is O(")-close to the

equilibrium (cycle) of the averaged system. This

torus is stable or unstable together with the

equilibrium (cycle) of the averaged system. For

Hamiltonian and reversible systems, the problem of

invariant manifolds is considered in the framework

of the KAM theory.

Averaging in Bogolyubov’s Systems

Systems in the standard form of Bogolyubov (1945)

are of the form

x ¼ "Xðt; x;"Þ; x 2 R

; 0 <" 1 ½10

It is assumed that the function X, besides the usual

smoothness conditions, satisfies the condition of

uniform average: the limit (time average)

ðxÞ¼ lim

T!1

Xðt; x; 0Þdt ½11

exists uniformly in x. The averaging principle of

Bogolyubov consists of the replacement of the

original system in standard form by the averaged

system

 ¼ " X

ðÞ½12

with a goal to provide an approximate description

of the behavior of x. This approach generalizes the

appro ach of the section ‘‘Averag ing principle’’ for

the case of constant frequencies (! = const). Upon

introducing in the given system with constant

frequencies the deviation from uniform rotation

 = ’  !t and denoting x = (I, ), we obtain a

system in the standard form [10]. Here the condition

of uniform average is fulfilled because X(t, x ,0) is a

quasiperiodic function of time t. The averag ed

system [12] for nonresonant frequencies coincides

with the averaged system [2]; for resonant frequen-

cies, it coincides with the partially averaged system

[3] (one should only supply systems [2] and [3] with

equations for some components of the vector ’  !t

that do not enter into the right-hand side of the

averaged system).

The averaging principle of Bogolyubov is justified

by three Bogolyubov theorems. According to the

first theorem, if (t ), 0  t  K=", is a solution of

the averaged system, and x(t) is a solution of the

exact system with initial condition x(0) = (0), then

for any >0 there exists "

() > 0 such that

jx(t )  (t)j < for 0  t  K=" and 0 <"<"

().

The second and the third Bogolyubov theorems

describe the motion in the neighborhoods of

equilibria and the limit cycles of the averag ed

system. In particular, if for an equilibrium real

parts of all its eigenvalues are differe nt from 0, or,

for a limit cycle, the absolute values of all but one

multipliers are different from 1, then the exact

system has a solution which eternally stays near

this equilibrium (cycle). The stability properties of

this solution are the same as the stability properties

of the corresponding equilibrium (cycle) of the

averaged system.

For systems of the form [10] a procedure exists

that, similarly to the procedure in the section

‘‘Elim ination of fast variab les, decoupl ing of slow

and fast motion s,’’ allows us to elimina te time t

from the right-hand side of the system with an

accuracy of the order of any power in " by means of

a transformation of variables. (To perform this

procedure, one should assume that the conditions

of uniform average are satisfied for functions

that arise in the process of constructing higher

approximations in this procedure ( Bogolyubuv and

Mitropol’skii 1961).) In the first approximation,

such a transformation of variables transforms the

original system into the averaged one.

The condition of uniform average is very impor-

tant for theory. If the limit in [11] exists, but

convergence is nonuniform in x, then the time

average X

could be, for example, a discontinuous

function of x, and the averaged system would not be

well defined.

Averaging in Slow–Fast Systems

Systems of the form [1] are particular cases of the

systems of the form

x ¼ f ðx; y;"Þ;

y ¼ "gðx; y;"Þ½13

which are called ‘‘slow–fast systems’’ (or systems

with slow and fast motions, with slow and fast

variables). The generalization of the approach of the

section ‘‘Avera ging principl e’’ for these systems is

the following averaging principle of Anosov (1960).

In the system [6], let x 2 M, y 2 R

, where M is a

smooth compact m-dimensional manifold . At " = 0,

the system for fast variables x contains slow

variables y as parameters. Assume that this system

(which is called ‘‘fast system’’) has a finite smooth

230 Averaging Methods

invariant measure 

and is ergodic for almost all

values of y. Introduce the averaged syst em

Y ¼ " GðYÞ; GðYÞ¼



ðMÞ

gðx; Y; 0Þd

According to the averaging principle, one should use

the solution Y(t) of the averaged system with initial

condition Y(0) = y(0) for approximate description of

slow motion y(t) in the original system. This

averaging principle is justified by the following

Anosov theorem [1]: for any positive  the measure

of the set E(, ") of initial data (from a compact in

the phase space) such that

max

0 t 1="

jyðtÞYðt Þj >

tends to 0 as " ! 0.

The particular case when the original system is

a Hamiltonian system depending on slowly vary-

ing parameter  = "t, and for almost all values of

 the motion o f the system with  = const is

ergodic on almost all energy levels, is considered

in Kasuga (1961).

For the case when the has strong mixing proper-

ties, see Bakhtin (2004) and Kifer (2004).

For slow–fast systems, there is also a general-

ization of approach of the previous section that uses

time averaging and the condition of uniform average

(Volosov 1962).

Applications of the Averaging Method

The averaging method is one of the most pro ductive

methods of perturbation theory, and its applications

are immense. It is widely used in celestial mechanics

and space flight dynamics for the description of the

evolution of motions of celestial bodies, in plasma

physics and theory of accelerators for description of

motion of charged pa rticles, and in radio engineer-

ing for the description of nonlinear oscillatory

regimes. There are also applications in hydrody-

namics, physics of lasers, optics, acoust ics, etc. (see

Arnol’d et al. (1988), Bogolyubov and Mitropol’skii

(1961), Lochak and Meunier (1988), Mitropol’skii

(1971), and Volosov (1962)).

See also: Central Manifolds, Normal Forms;

Diagrammatic Techniques in Perturbation Theory;

Hamiltonian Systems: Stability and Instability Theory;

KAM Theory and Celestial Mechanics; Multiscale

Approaches; Random Walks in Random Environments;

Separatrix Splitting; Stability Problems in Celestial

Mechanics; Stability Theory and KAM.

Further Reading

Anosov DV (1960) Averaging in systems of ordinary differential

equations with rapidly oscillating solutions. Izvestiya Akade-

mii Nauk SSSR, Ser. Mat. 24(5): 721–742 (Russian).

Arnol’d VI (1983) Geometrical Methods in the Theory

of Ordinary Differential Equations. New York–Berlin:

Springer.

Arnol’d VI, Kozlov VV, and Neishtadt AI (1988) Mathematical

Aspects of Classical and Celestial Mechanics, Encyclopaedia

of Mathematical Sciences, vol. 3. Berlin: Springer.

Bakhtin VI (2004) Crame´r asymptotics in the averaging method

for systems with fast hyperbolic motions. Proceedings of the

Steklov Institute of Mathematics 244(1): 79.

Bogolyubov NN (1945) On some statistical methods in mathe-

matical physics. Akad. Nauk USSR. L’vov (Russian).

Bogolyubov NN and Mitropol’skii YuA (1961) Asymptotic

Methods in the Theory of Nonlinear Oscillations. New York:

Gordon and Breach.

Giacaglia GEO (1972) Perturbation Methods in Nonlinear

Systems, Applied Mathematical Science, vol. 8. Berlin: Springer.

Kasuga T (1961) On the adiabatic theorem for the

Hamiltonian system of differential equations in the classical

mechanics I, II, III. Proceedings of the Japan Academy 37(7):

366–382.

Kevorkian J and Cole JD (1996) Multiple Scale and Singular

Perturbations Methods, Applied Mathematical Sciences,

vol. 114. New York: Springer.

Kifer Y (2004) Some recent advances in averaging. In: Modern

Dynamical Systems and Applications, 403. Cambridge:

Cambridge Unive rsity Press.

Lochak P and Meunier P (1988) Multiphase Averaging for

Classical Systems, Applied Mathematical Sciences, vol. 72.

New York: Springer.

Mitropol’skii YuA (1971) Averaging Method in Nonlinear

Mechanics. Kiev: Naukova Dumka (Russian).

Poincare´ H (1957) Les Me´thodes Nouvelles de la Me´canique

Ce´leste, vols. 1–3. New York: Dover.

Sanders JA and Verhulst F (1985) Averaging Methods in

Nonlinear Dynamical Systems, Applied Mathematical

Sciences, vol. 59. New York: Springer.

Volosov VM (1962) Averaging in systems of ordinary differential

equations. Russian Mathematical Surveys 17(6): 1–126.

Averaging Methods 231

Axiomatic Approach to Topological Quantum Field Theory

C Blanchet, Universite

de Bretagne-Sud, Vannes,

France

V Turaev, IRMA, Strasbourg, France

Introduction

The idea of topological invariants defined via path

integrals was introduced by A S Schwartz (1977) in a

special case and by E Witten (1988) in its full

power. To formalize this idea, Witten (1988)

introduced a notion of a topological quantum field

theory (TQFT). Such theories, independent of

Riemannian metrics, are rather rare in quantum

physics. On the other hand, they admit a simple

axiomatic description first suggested by M Atiyah

(1989). This description was inspired by G Segal’s

(1988) axioms for a two-dimensional conformal

field theory. The axiomatic formulation of TQFTs

makes them suitable for a purely mathematical

research combining methods of topology, algebra,

and mathematical physics. Several authors explored

axiomatic foundations of TQFTs (see Quinn (1995)

and Turaev (1994).

Axioms of a TQFT

An (n þ 1)-dimensional TQFT (V, ) over a scalar

field k assigns to every closed oriented n-dimen-

sional manifold X a finite-dimensional vector space

V(X) over k and assigns to every cobordism

(M, X, Y)ak-linear map

ðMÞ¼ðM; X; YÞ : Vð XÞ!VðYÞ

Here a cobordism (M, X, Y) between X and Y is a

compact oriented (n þ 1)-dimensional manifold M

endowed with a diffeomorphism @M 

X q Y (the

overline indicates the orientation reversal). All

manifolds and cobordisms are supposed to be

smooth. A TQFT must satisfy the following axioms.

1. Naturality Any orientation-preserving diffeo-

morphism of closed oriented n-dimensional mani-

folds f : X !X

induces an isomorphism f

]

: V

(X)!V(X

). For a diffeomorphism g between the

cobordisms (M, X, Y)and(M

, X

, Y

), the follow-

ing diagram is commutative:

VðXÞ

!

ðg

]

VðX

ðMÞ

ðM

VðYÞ

!

ðg

]

VðY

2. Functoriality If a cobordism (W, X, Z )is

obtained by gluing two cobordisms (M, X, Y)and

, Y

, Z) along a diffeomorphism f : Y !Y

,then

the following diagram is commutative:

VðXÞ

!

ðWÞ

VðZÞ

ðMÞ

ðM

VðYÞ

!

]

VðY

3. Normalization For any n-dimensional manifold

X, the linear map

ð½0; 1XÞ : VðXÞ!VðXÞ

is identity.

4. Multiplicativity There are functorial

isomorphisms

VðX q YÞVðXÞVðYÞ

Vð;Þ  k

such that the following diagrams are commutative:

VððX q YÞqZÞ VðXÞVðYÞðÞVðZÞ

VðX qðY q ZÞÞ  VðXÞ VðYÞVðZÞðÞ

VðX q;Þ  VðXÞk

VðXÞ¼ VðXÞ

Here = 

is the tensor product over k. The

vertical maps are respectively the ones induced

by the obvious diffeomorphisms, and the stan-

dard isomorphisms of vector spaces.

5. Symmetry The isomorphism

VðX q YÞVðY q XÞ

induced by the obvious diffeomorphism corre-

sponds to the standard isomorphism of vector

spaces

VðXÞVðYÞVðYÞVðXÞ

Given a TQFT (V, ), we obtain an action of the

group of diffeomorphisms of a closed oriented

n-dimensional manifold X on the vector space

V(X). This action can be used to study this group.

An important feature of a TQFT (V, ) is that it

provides numerical invariants of compact oriented

(n þ 1)-dimensional manifolds without boundary.

Indeed, such a manifold M can be considered as a

cobordism between two copies of ; so that (M) 2

Hom

(k, k) = k. Any compact oriented (n þ 1)-

dimensional manifold M can be considered as a

232 Axiomatic Approach to Topological Quantum Field Theory

cobordism between ; and @M; the TQFT assigns to

this cobordism a vector (M) in Hom

(k,

V(@M)) = V(@M) called the vacuum vector.

The manifold [0, 1]  X, considered as a cobord-

ism from

X q X to ; induces a nonsingular pairing

Vð

XÞVðXÞ!k

We obtain a functorial isomorphism V(

X) =

V(X)



= Hom

(V(X), k).

We now outline definitions of several important

classes of TQFTs.

If the scalar field k has a conjugation and all the

vector spaces V(X) are equipped with natural

nondegenerate Hermitian forms, then the TQFT

(V, ) is Hermitian. If k = C is the field of complex

numbers and the Hermitian forms are positive

definite, then the TQFT is unitary.

A TQFT (V, ) is nondegenerate or cobordism

generated if for any closed oriented n-dimensional

manifold X, the vector space V(X) is generated by

the vacuum vectors derived as above from the

manifolds bounded by X.

Fix a Dedekind domain D  C. A TQFT (V, )

over C is almost D-integral if it is nondegenerate and

there is d 2 C such that d(M) 2 D for all M with

@M = ;. Given an almost integral TQFT (V, )anda

closed oriented n-dimensional manifold X,wedefine

S(X)tobetheD-submodule of V(X) generated by all

the vacuum vectors. This module is preserved under

the action of self-diffeomorphisms of X and yields a

finer ‘‘arithmetic’’ version of V(X).

The notion of an (n þ 1)-dimensional TQFT over

k can be reformulated in the categorical language as

a symmetric monoidal functor from the category of

n-manifolds and (n þ 1)-cobordisms to the category

of finite-dimensional vector spaces over k. The

source category is called the (n þ 1)-dimensional

cobordism category. Its objects are closed oriented

n-dimensional manifolds. Its morphisms are cobord-

isms considered up to the following equivalence:

cobordisms (M, X, Y) and (M

, X, Y) are equivalent

if there is a diffeomorphism M !M

compatible

with the diffeomorphisms @M 

X q Y  @M

TQFTs in Low Dimensions

TQFTs in dimension 0 þ 1 = 1 are in one-to-one

correspondence with finite-dimensional vector

spaces. The correspondence goes by associating

with a one-dimensional TQFT (V, ) the vector

space V(pt) where pt is a point with positive

orientation.

Let (V, ) be a two-dimensional TQFT. The linear

map  associated with a pair of pants (a 2-disk with

two holes considered as a cobordism between two

circles S

q S

and one circle S

) defines a commu-

tative multiplication on the vector space A= V(S

The 2-disk, considered as a cobordism between S

and ;, induces a nondegenerate trace on the algebra

A. This makes A into a commutative Frobenius

algebra (also called a symmetric algebra). This

algebra completely determines the TQFT (V, ).

Moreover, this construction defines a one-to-one

correspondence between equivalence classes of two-

dimensional TQFTs and isomorphism classes of

finite dimensional commutative Frobenius algebras

(Kock 2003).

The formalism of TQFTs was to a great extent

motivated by the three-dimensional case, specifi-

cally, Witten’s Chern–Simons TQFTs. A mathema-

tical definition of these TQFTs was first given

by Reshetikhin and Turaev using the theory of

quantum groups. The Witten–Reshetikhin–Turaev

three-dimensional TQFTs do not satisfy exactly the

definition above: the naturality and the functoriality

axioms only hold up to invertible scalar factors

called framing anomalies. Such TQFTs are said to

be projective. In order to get rid of the framing

anomalies, one has to add extra structures on the

three-dimensional cobordism category. Usually one

endows surfaces X with Lagrangians (maximal

isotropic subspaces in H

(X; R)). For 3-cobordisms,

several competing – but essentially equivalent –

additional structures are considered in the literature:

2-framings (Atiyah 1989), p

-structures (Blanchet

et al. 1995), numerical weights (K Walker, V Turaev).

Large families of three-dimensional TQFTs are

obtained from the so-called modular categories.

The latter are constructed from quantum groups at

roots of unity or from the skein theory of links.

See Quantum 3-Manifold Invariants.

Additional Structures

The axiomatic definition of a TQFT extends in

various directions. In dimension 2 it is interesting to

consider the so-called open–closed theories involving

1-manifolds formed by circles and intervals and

two-dimensional cobordisms with boundary

(G Moore, G Segal). In dimension 3 one often

considers cobordisms including framed links and

graphs whose components (resp. edges) are labeled

with objects of a certain fixed category C.Insucha

theory, surfaces are endowed with finite sets of

points labeled with objects of C and enriched with

tangent directions. In all dimensions one can study

manifolds and cobordisms endowed with homotopy

classes of mappings to a fixed space (homotopy

quantum field theory, in the sense of Turaev).

Additional structures on the tangent bundles – spin

Axiomatic Approach to Topological Quantum Field Theory 233

structures, framings, etc. – may be also considered

provided the gluing is well defined.

See also: Braided and Modular Tensor Categories; Hopf

Algebras and q-Deformation Quantum Groups; Indefinite

Metric; Quantum 3-Manifold Invariants; Topological

Gravity, Two-Dimensional; Topological Quantum Field

Theory: Overview.