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

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theory, which are useful when working with Ka¨ hler

or hyper-Ka¨hler manifolds.

In physics, moduli spaces are often described as

symplectic reductions of infinite-dimensional sym-

plectic manifold s by infinite-dimensional groups

(although the moduli spaces themselves are usually

finite-dimensional). One example is given by moduli

spaces of holomorphic vector bundles, which

can also be described using Yang–Mills theory

(cf. Atiyah and Bott (1982)).

The Yang–Mills equations arose in physics as

generalizations of Maxwell’s equations. They have

become important in differential and algebraic

geometry formulated over arbitrary compact oriented

Riemannian manifolds, and in particular over com-

pact Riemann surfaces and higher dimensional Ka¨ hler

manifolds. The fundamental theorem of Donaldson,

Uhlenbeck, and Yau that a holomorphic bundle over

acompactKa¨hler manifold admits an irreducible

Hermitian Yang–Mills connection if and only if it is

stable can be thought of as an infinite-dimensional

illustration of the link between symplectic reduction

and geometric invariant theory.

Let M be a compact oriented Riemannian mani-

fold and let E be a fixed complex vector bundle over

M with a Hermitian metric. Recall that a connection

A on E (or equivalently on its frame bundle) can be

defined by a covariant derivative d

: 

(E) !



pþ1

(E), where 

(E) denotes the space of

-sections of



M  E (i.e., the space of

p-forms on M with value s in E). This covariant

derivative satisfies the extended Leibniz rule

ð ^ Þ¼ðd

Þ^ þð1Þ

 ^ d



for  2 

(E),  2 

(E), and therefore is deter-

mined by its restriction d

: 

(E) ! 

(E). The

Leibniz rule implies that the difference of two

connections is given by an E  E



-valued 1-form

on M, and hence that the space of all connections on

E is an infinite- dimensional affine space A based on

the vector space 

(E  E



). Similarly, the space of

all unitary connections on E (i.e., connections

compatible with the Hermitian metric on E)isan

infinite-dimensional affine space based on the space

of 1-forms with values in the bundle g

of skew-

adjoint endomorphisms of E. The Leibniz rule also

implies that the composition d

 d

: 

(E) !



(E) commutes with multiplication by smooth

functions, and thus we have

 d

ðsÞ¼F

for all C

sections s of E, where F

2 

)is

defined to be the curvature of the unitary connection

A. The Yang–Mills functional on the space A of all

unitary connections on E is defined as the L

-norm

square of the curvature, given by the integral over M

of the product of the function kF

and the volume

form on M defined by the Riemannian metric and the

orientation. The Yang–Mills equations are the Euler–

Lagrange equations for this functional, given by

 F

¼ 0

where d

has been extende d in a natural way to





). The gauge group G, that is, the group of

unitary automorphisms of E, prese rves the Yang–

Mills functional and the Yang–Mills equations.

If M is a complex manifold, we can identify the

space A

(1, 1)

of unitary connections on E with

curvature of type (1,1) with the space of holomorphic

structures on E, by associating to a holomorphic

structure E the unitary connection whose (0, 1)-

component is given by the



@-operator defined by E.

This space A

(1, 1)

is an infinite-dimensional complex

subvariety of the infinite-dimensional complex affine

space A, acted on by the complexified gauge group

(the group of complex C

automorphisms of E),

and two holomorphic structures are isomorphic if

and only if they lie in the same G

-orbit.

When (M, !) is a compact Ka¨ hler manifol d, there

is a G-invariant Ka¨hler form  on A defined by

ð; Þ¼

8

trð ^ Þ^!

n1

where n is the complex dimension of M. The Lie

algebra of G is the space 

) of sections of g

and there is a moment map  : A!(

))



for

the action of G on A given by the composition of

A 7!

8

^ !

n1

2 

ðg

with integration over M.OnA

(1, 1)

the norm square

of this moment map agrees up to a constant factor

with the Yang–Mills functional, which is minimized

by the Hermitian Yang–Mills connections.

As in the finite-dimensional situation, for a suitable

definition of stability, the moduli space of stable

holomorphic bundles of topological type E over M

(which plays the role of the geometric invariant

theory quotient) can be identified with the moduli

space of (irreducible) Hermitian Yang–Mills connec-

tions on E (which plays the roˆ le of the symplectic

reduction). This was proved in general for vector

bundles over compact Ka¨ hler manifolds Uhlenbeck

and Yau with a different proof for nonsingular

complex projective varieties given by Donaldson.

Over a compact Riemann surface M the situation is

relatively simple, as all connections on E have

curvature of type (1, 1) and so the infinite-dimensional

Moduli Spaces: An Introduction 457

complex affine space A can be identified with the

space C of holomorphic structures on E. A moment

map for the action of the gauge group on A is given by

assigning to a connection A 2Aits curvature F



), and, after a suitable central constant has been

added, the Hermitian Yang–Mills connections are

exactly the zeros of the moment map.

A holomorphic bundle E over a Riemann surface

M is stable (respectively semistable) if (F) <(E)

(respectively (F)  (E)) for every proper sub-

bundle F of E, where

ðFÞ ¼ degðFÞ=rankðFÞ

When the theory of stability of holomorphic vector

bundles was first introduced, Narasimhan and

Seshadri proved that a holomorphic vector bundle

over M is stable if and only if it arises from an

irreducible representation of a certain central exten-

sion of the fundamental group 

(M). Atiyah and

Bott (1982) translated this in terms of connections to

show that a holomorphic vector bundle over M is

stable if and only if it admits a unitary connection

with constant central curvature. They deduced from

this the existence of a homeomorphism between the

moduli space M(n, d) of stable bundles of rank n and

degree d over M and the moduli space of irreducible

connections with constant central curvature on a

fixed C

bundle E of rank n and degree d over M.

See also: BF Theories; Calibrated Geometry and Special

Lagrangian Submanifolds; Cohomology Theories; Floer

Homology; Gauge Theoretic Invariants of 4-Manifolds;

Gauge Theory: Mathematical Applications; Geometric

Measure Theory; Geometric Phases; Hamiltonian Group

Actions; Instantons: Topological Aspects; Intersection

Theory; Riemann Surfaces; Several Complex Variables:

Basic Geometric Theory; Several Complex Variables:

Compact Manifolds; Topological Gravity, Two-

Dimensional; WDVV Equations and Frobenius Manifolds.

Further Reading

Atiyah MF and Bott R (1982) The Yang–Mills equations over

Riemann surfaces. Philosophical Transactions of the Royal Society

London 308: 523–615.

Cox D and Katz S (1999) Mirror Symmetry and Algebraic

Geometry. Math Surveys and Monographs, vol. 68.

Providence, RI: American Mathematical Society.

Deligne P and Mumford D (1969) The irreducibility of the space

of curves of given genus. Publications of the Institute Hautes

‘Etudes Scientifique’ 36: 75–110.

Dijkgraaf R et al. (eds.) (1995) The Moduli Space of Curves.

Boston, MA: Birkha¨user.

Donaldson SK (1984) Instantons and geometric invariant theory.

Communications in Mathematical Physics 93: 453–460.

Fulton W and Pandharipande R (1997) Notes on stable maps and

quantum cohomology. In: Algebraic Geometry, Santa Cruz

1995, Proceeding of the Symposia in Pure Mathematics 62

vol. 2 (1997), 45–96.

Gieseker D (1983) Geometric invariant theory and applications to

moduli problems. In: Invariant Theory (Montecatini, 1982),

Lecture Notes in Mathematics vol. 996, pp. 45–73. Berlin:

Springer.

Griffiths P (1984) Topics in transcendental algebraic geometry.

Annals of Mathematical Studies,vol.106.Princeton:Princeton

University Press.

Harris J and Morrison I (1998) Moduli of Curves. Graduate

Texts in Mathematics, vol. 107. Berlin: Springer.

Keel S and Mori S (1997) Quotients by groupoids. Annals of

Mathematics 145(2): 193–213.

Kirwan FC (1985) Cohomology of Quotients in Algebraic and

Symplectic Geometry. Math. Notes. vol. 31. Princeton: Princeton

University Press.

Kolla´r J (1997) Quotient spaces modulo algebraic groups. Annals

of Mathematics 145(2): 33–79.

Kolla´r J and Mori S (1998) Birational Geometry of Algebraic

Varieties. Cambridge Tracts in Mathematics, vol. 134. Cam-

bridge: Cambridge University Press.

Lehto O (1987) Univalent Functions and Teichmu¨ ller Spaces.

Graduate Texts in Mathematics. vol. 109. Berlin: Springer.

Mori S (1987) Classification of higher dimensional varieties,

Algebraic Geometry, Bowdoin 1985. Proceedings of the

Symposia in Pure Mathematics 46: 269–331.

Mumford D (1965) Geometric Invariant Theory, 3rd edn. with

Fogarty J and Kirwin F (1994). Berlin: Springer.

Mumford D (1977) Stability of projective varieties. L’Enseigne-

ment Mathe´matiques 23: 33–110.

Newstead PE (1978) Introduction to Moduli Problems and Orbit

Spaces. Tata Institute Lecture Notes. Berlin: Springer.

Popp H (1977) Moduli Theory and Classification Theory of

Algebraic Varieties. Springer Lecture Notes in Mathematics,

vol. 620. Berlin: Springer.

Seshadri C (1975) Theory of moduli, Proceedings of the Symposia

in Pure Mathematics vol. 29, Algebraic Geometry. American

Mathematical Society.

Sundaramanan D (1980) Moduli, Deformations and Classifica-

tions of Compact Complex Manifolds. Pitman Research Notes

in Mathematics, vol. 45. Harlow: Longman.

Verdier J-L and Le Potier J (1985) Module des fibre´s stables sur

les courbes alge´briques. Progress in Mathematics, vol. 54.

Boston, MA: Birkha¨user.

Viehweg E (1995) Quasi-Projective Moduli for Polarized Mani-

folds. Berlin: Springer.

Multicomponent Fluids see Interfaces and Multicomponent Fluids

458 Moduli Spaces: An Introduction

Multi-Hamiltonian Systems

F Magri, Universita

di Milano Bicocca, Milan, Italy

MPedroni, Universita

di Bergamo, Dalmine (BG), Italy

Introduction

Since the late 1970s, a particular attention in the

theory of integrability has been payed to systems

admitting more than one Hamiltonian representa-

tion. The first examples belonged to the class of

infinite-dimensional systems (i.e., partial differen-

tial equations), like the Korteweg–de Vries (KdV)

equation, the Ablowitz–Kaup–Newell–Segur

system, and many other soliton equations (see

Bi-Hamiltonian Methods in Soliton Theory). It

was realized soon that finite-dimensional integr-

able systems are also likely to possess a

bi-Hamiltonian representation. Moreover, a geo-

metric setting for the study of bi-Hamiltonian

systems was established, with the introduction of

the so-called bi-Hamiltonian manifolds. They are

Poisson manifolds with an additional Poisson

structure, fulfilling a suitable compatibility con-

dition with the initial Poisson bracket. An

important program for the study and the classi-

fication of (finite-dimensional) bi-Hamiltonian

manifolds was started in the 1990s by Gelfand

and Zakharevich. They pointed out that the

geometry of such manifolds is extremely rich

and complicated.

Inthisarticlewepresentthebasicfacts

concerning the bi-Hamiltonian geometry and its

relations with the theory of integrable systems,

referring to Recursion Operators in Classical

Mechanics in this encyclopedia for the connections

with separable systems of Jacobi. In the first

section we give the definitions of bi-Hamiltonian

manifold and bi-Hamiltonian system, and we

present some properties of the former. The next

section contains three concrete examples (the Euler

top, the open Toda lattice, and a stationary KdV

flow) and two important classes of bi-Hamiltonian

manifolds, both related to Lie algebras. This is

followed by a discussion of the iterative construc-

tion of first integrals in involution for a given

bi-Hamiltonian system. This procedure is particu-

larly efficient in the case of Poisson–Nijenhuis

manifolds, that is, those bi-Hamiltonian manifolds

whose second Poisson structure can be obtained by

composing the first one with a suitable recursion

operator.

Bi-Hamiltonian Systems

First of all, we recall some fundamental definitions

from the theory of Poisson manifolds, which are the

natural setting for the study of Hamiltonian systems.

Let M be a finite-dimensional C

-differentiable

manifold and let C

(M) be the space of C

functions from M to R. A Poisson bracket on M is

a skew-symmetric R-bilinear map

f; g:C

ðMÞC

ðMÞ!C

ðMÞ

fulfilling the Jacobi identity

ffF; Gg; HgþffH; Fg; GgþffG; Hg; Fg¼0

and the Leibniz rule

fFG; Hg¼FfG; HgþfF; HgG

A Poisson manifold is a differentiable manifold

endowed with a Poisson bracket. Starting from a

Poisson bracket, one can introduce a tensor field P

of type (2, 0), which we consider as a map from



M to TM, defined by

hdG; P dFi¼fF; Gg

or, using coordinates on M,byP

= {x

, x

}. This

tensor field is called the Poisson tensor associated

with { , }. It is skew-symmetric, and its components

satisfy the cyclic condition

þ P

@ x

þ P

¼ 0

meaning that the Schouten bracket [P, P] vanishes.

On a Poisson manifold, the vector field

= {H, } = P dH is called the Hamiltonian

vector field associated with H. In coordinates,

= P

@H=@x

. The Jacobi identity is equivalent

to the statement that the map H 7! X

, assigning

to a function H its Hamiltonian vector field X

,is

a Lie algebra homomorphism:

fF;Gg

¼½X

; X

½1

A Casimir function is a function H such that

= 0, that is, a function which is in involution

with any other function on M. In terms of the

Poisson tensor, a Casimir is a function whose

differential belongs to the kernel of P .

The most famous class of Poisson manifolds is

certainly that of symplectic manifolds. They can be

seen as nondegenerate Poisson manifolds. Indeed, if

a Poisson tensor P is invertible, then its inverse

defines a closed nondegenerate 2-form (i.e., a

symplectic form). Moreover, any Poisson manifold

turns out to be foliated in symplectic leaves.

Multi-Hamiltonian Systems 459

Let us introduce now the bi-Hamiltonian manifolds,

which can be considered as a geometric setting for the

study of integrable Hamiltonian systems. A manifold

M endowed with two Poisson brackets, {, }and{, }

is said to be bi-Hamiltonian if the brackets are

compatible, that is, if any linear combination (with

constant coefficients) of them is still a Poisson bracket.

Such a linear combination automatically satisfies all

properties of a Poisson bracket except the Jacobi

identity. This is fulfilled if and only if the following

compatibility condition holds:

fF; fG; Hgg

þfH; fF; Ggg

þfG; fH; Fgg

þfF; fG; Hg

gþfH; fF; Gg

þfG; fH; Fg

g¼0 ½2

for any triple (F, G, H) of functions on M. This

amounts to saying that the sum of the two Poisson

brackets is also a Poisson bracket. In this case the

two (compatible) Poisson brackets are said to form a

Poisson pair.

There are some interesting equivalent forms of the

compatibility condition [2]. First of all, in terms of

the components of the Poisson tensors P and P

,it

reads

@ðP

þ P

@ðP

þ P

@ðP

þðP

¼ 0

that is, the Schouten bracket [P, P

] vanishes. More-

over, if X

= P dF is the Hamiltonian vector field

associated with F 2 C

(M) by means of P and

= P

dF is the one obtained by P

, the compat-

ibility condition takes the form

½X

; Y

þ½Y

; X

¼X

fF;Gg

þ Y

fF;Gg

8F; G 2 C

ðMÞ½3

to be compared with [1]. Moreover, in terms of Lie

derivatives we have the equivalent condition

þ L

P ¼ 0 8F 2 C

ðMÞ½4

Now we turn our attention to special vector fields

that can be selected on a bi-Hamiltonian manifold

M. Let P and P

be the Poisson tensors associated

with the (compatible) Poisson brackets of M.A

vector field X on M is said to be bi-Hamiltonian if it

is Hamiltonian with respect to both Poisson struc-

tures, that is, if there exist two functions H

and H

such that

X ¼ P dH

¼ P

½5

We will see in the following that such vector fields

are likely to have a number of first integrals in

involution, and thus they are good candidates for a

geometric description of integrable systems. The next

section is devoted to examples of bi-Hamiltonian

(and multi-Hamiltonian) systems.

Examples

The first example is the Euler top, that is, free

motions of a rigid body with a fixed point. The

equations of motion are



 I



and its cyclic permutations. They define a vector

field in R

, which is well known to be Hamiltonian

with respect to the Lie–Poisson structure on the

(dual of the) Lie algebra of 3  3 skew-symmetric

matrices. This means that



¼fH; 

g; j ¼ 1; 2; 3

where

H ¼





is the kinetic energy and the bracket { , } is defined

by {

, 

} =

and its cyclic permutations. Another

Hamiltonian representation is given by



¼fK; 

; j ¼ 1; 2; 3

where

K ¼



þ 



and the new bracket { , }

is defined by {

, 

}



and its cyclic permutations.Any linear

combination of the two brackets has the form of

the second one, and it is very easy to show that the

Jacobi identity is satisfied for such a bracket.

Therefore, the Euler top is a bi-Hamiltonian system.

Let us also notice that

fK; 

g¼fH; 

¼ 0; j ¼ 1; 2; 3

that is, K is a Casimir function for the Lie–Poisson

bracket and H is a Casimir function for the new

Poisson bracket. Hence, we have the following

(recursion) relations:

fK; 

g¼0

fH; 

g¼fK; 

0 ¼fH; 

½6

From a geometrical point of view, the situation is as

follows. The symplectic leaves of { , } are the level

surfaces of K, that is, spheres, while the symplectic

460 Multi-Hamiltonian Systems

leaves of { , }

are the ellipsoids H = constant. Their

intersections are Lagrangian submanifolds for both

symplectic leaves (in the compact case they are the

Arnol’d–Liouville tori of the integrable systems, that

in this case coincide with the trajectories).

Let us consider now the (three-particle) open

Toda lattice. It consists in three particles (with

masses equal to 1) moving on the line under a

nearest-neighbor interaction of exponential type.

The Hamiltonian is given by

H ¼

þ p



þ expðq

q

Þþexpðq

q

and the system is of course Hamiltonian with

respect to the canonical Poisson structure of R

P ¼

000100

000010

0000 01

1000 0 0

0100 0 0

0010 0 0

But the Toda vector field can also be written as

dK, where K = p

þ p

is the total momen-

tum and

011 p

10 1 0 p

1 10 0 0 p

00 0 e

ðq

q

0 p

0 e

ðq

q

ðq

q

00p

0 e

ðq

q

is a Poisson tensor, which turns out to be compatible

with P. The generalization to an arbitrary number of

particles is straightforward. Hence, the open Toda

lattice is a bi-Hamiltonian system. In the next section

we will show that this property can be used to

construct a maximal set of integrals of motion for the

Toda lattice, which are automatically in involution.

The third example – a stationary reduction of the

KdV equation – comes from the field of soliton

equations. Let us recall that the first members of the

KdV hierarchy are

¼ u

ðu

xxx

 6uu

ÞðKdV equationÞ

xxxxx

 10uu

xxx

20u

þ 30u



½7

It is well known how to find finite-dimensional

reductions for the KdV equation, giving rise to explicit

solutions. Indeed, the set of singular points of a given

vector field of the hierarchy is a finite-dimensional

manifold which is invariant under the flows of the

other vector fields, due to the fact that the flows

commute. The (finite-dimensional) systems obtained

by restricting the KdV hierarchy to such invariant

manifolds are called the stationary reductions of KdV.

Let us consider explicitly the reduction corresponding

to the third vector field of the hierarchy. The set of its

critical points is given by

xxxxx

 10uu

xxx

 20u

þ 30u

¼ 0 ½8

and its dimension is 5, since we can use the values of

u, u

, u

xxx

, and u

xxxx

at a fixed point x

(i.e., the

Cauchy data) as global coordinates. For the sake of

simplicity, we set

¼ uðx

Þ; u

¼ u

ðx

Þ; u

¼ u

ðx

¼ u

xxx

ðx

Þ; u

¼ u

xxxx

ðx

In order to compute the reduced equations of the

first flow of [7], we have to take its x-derivative and

to use the constraint [8] and its differential

consequences to eliminate all the derivatives of

order higher than 4. We obtain the equations

¼ u

;

¼ u

;

¼ u

;

¼ u

¼ 10u

þ 20u

 30u

½9

In the same way, for the KdV equation we get

ðu

 6u



þ 2u

 30u



þ 6u

þ 2u



30u

 60u



10u

þ 10u



100u

 60u

 120u



½10

There are two compatible Poisson structures giving

a bi-Hamiltonian formulation of both systems. The

corresponding Poisson tensors are

P ¼

00 0 2 0

00 20 20u

02 0 20u

20u

2020u

0 140u

20u

020u

20u

140u

þ20u

Multi-Hamiltonian Systems 461

and

03u



0 3u

3u

4u

 15u

03u

0 u

þ 15u

þ 30u

3u

u

 15u

 40u

30u

 60u

6u

þ 15u

u

 30u

u

þ 40u



30u

þ 60u

In fact, if we call X

and X

the vector fields given

by [9] and [10], then the following recursion

relations hold:

P dH

¼0

¼ P dH

¼P

¼ P dH

¼P

0 ¼P

½11

where

¼u

þ 10u

þ 5u

 10u

 2u

þ u

 20u

þ 15u



 6u

 u

þ 12u



16u

 12u

þ 60u

 36u



Therefore, the vector fields X

and X

are

bi-Hamiltonian. The geometry of this bi-Hamiltonian

manifold is similar to the one of the first example. The

symplectic leaves of both Poisson structures

have dimension 4, and the Lagrangian foliation

(given by the level submanifolds of H

, H

,andH

)

is contained in the intersections of such leaves. This

Lagrangian foliation is called by Gelfand and Zakhar-

evich the ‘‘axis’’ of the bi-Hamiltonian manifold.

We also notice that the relations [11] can be

collected in the statement that the function

H() = H



þ H

 þ H

is a Casimir of the Poisson

pencil P



= P

 P, that is,



dHðÞ¼0

The importance of the stationary reductions of

the KdV hierarchy lies in the fact that (as noticed

in the early works on the subject) the reduced

equations can be solved by means of the classical

method of separation of variables. We mention

that the separability of these systems is a par-

ticular instance of a general result, which is

valid for quite a wide class of bi-Hamiltonian

manifolds.

Next we present an important class of

bi-Hamiltonian manifolds. We recall that the

dual g



of a finite-dimensional Lie algebra g

possesses a canonical Poisson structure, called the

Lie–Poisson structure. It is defined as

fF; GgðXÞ¼hX; ½ dFðXÞ; dGðXÞi ½12

where F, G 2 C



) and their differentials at X 2 g



are seen as elements of g.IfX

is a fixed element in



, the constant Poisson bracket

fF; Gg

ðXÞ¼hX

; ½dFðXÞ; dGðXÞi ½13

is compatible with the Lie–Poisson bracket. In fact, the

Poisson pencil { , }



= { , }  { , }

is obtained from

{ , } by applying the translation X 7! X þ X

;

hence, it is a Poisson bracket for every value of the

constant . The method of translation of the argument,

due to Manakov, provides a lot of bi-Hamiltonian

vector fields for this bi-Hamiltonian manifold. One has

to consider an Ad



-invariant function on g



, that is, a

function H 2 C



)suchthat

hX; ½dH ðX Þ; xi ¼ 0 8 x 2 g; X 2 g



It is clearly a Casimir function for the Lie–Poisson

bracket, and this implies that the function

X 7!H(X  X

) is a Casimir of the Poisson pencil.

If this function can be developed as a Laurent series

in , its coefficients H

fulfill the recursion relations

jþ1

; 



¼fH

; g

½14

and thus give rise to a sequence of bi-Hamiltonian

vector fields.

The last example is a generalization of the

previous one. For the sake of simplicity, we consider

a Lie algebra g of matrices such that the trace of the

product is nondegenerate, and the space M = g

g  g.IfF 2 C

(M), its differential at a point

, x

) can be identified with the element (@F=@x

@F=@x

)ofM given by

j¼0

Fðx

þ v

; x

þ v

Þ¼tr



462 Multi-Hamiltonian Systems

for all v

, v

2 g. The manifold M has a three-

dimensional family of pairwise compatible Poisson

brackets:

fF; Gg

ðx

; x

Þ¼tr x

;



fF; Gg

; x

ðÞ¼tr x

;



fF; Gg

ðx

; x

Þ¼tr x

;



þx

;



;



Notice that the first two brackets restrict to the

submanifolds x

= constant and give rise to the

bi-Hamiltonian structure presented in the previous

example (via the identification between g and g



given by the trace of the product). This example can

be generalized to an arbitrary number n of copies of

g. In this case there is an (n þ 1)-dimensional family

of pairwise compatible Poisson brackets, which can

be shown to be Lie–Poisson brackets with respect to

suitable Lie algebra structures on g

. According to

Reyman and Semenov–Tian–Shansky, these brackets

can also be casted in the R-matrix formalism.

Also in this case, the Ad-invariant functions on g

give rise to functions in involution on our multi-

Hamiltonian manifold. For example, if H

()

denotes

the 

-coefficient of tr(x

 þ x

)



, then the recur-

sion relations

ðÞ

; g

¼fH

ðÞ

kþ1

; g

l þ1

; k  0; l ¼ 0; 1

hold, and they imply the existence of tri-Hamiltonian

vector fields on M.

Finally, we mention that the bi-Hamiltonian

structure of the stationary flow of KdV – discussed

above – can be obtained as a suitable reduction of

the multi-Hamiltonian structure on g

, where

g = sl(2, R). A similar statement holds for the other

stationary flows of the Gelfand–Dickey hierarchies.

Iterative Properties and Integrability

In this section we show how to use the bi-

Hamiltonian formulation of a given system to explain

its integrability. In the cases similar to the open Toda

lattice, where one of the Poisson structures is

nondegenerate, one can introduce a recursion opera-

tor and employ its powers in order to generate a

chain of integrals of motion in involution. In the

other examples, where the bi-Hamiltonian structure

is degenerate, the conserved quantities turn out to be

the coefficients of Casimir functions of the Poisson

pencil.

If (M,{ , }, { , }

) is a bi-Hamiltonian manifold,

we call bi-Hamiltonian hierarchy a sequence {H

}

k0

of functions on M fulfilling the recursion relations

f; H

kþ1

g¼f; H

; k  0 ½15

In terms of Poisson tensors we have that

P dH

kþ1

= P

. A bi-Hamiltonian hierarchy clearly

gives rise to an infinite sequence of bi-Hamiltonian

vector fields,

¼ P dH

¼ P

k1

; k  1 ½16

The functions H

are in involution with respect to

both Poisson brackets. Indeed, for k > j, one has

; H

g¼fH

; H

k1

¼fH

jþ1

; H

k1

g¼

¼fH

; H

so that {H

, H

} = 0 for all j, k  0, and therefore

, H

}

= 0 for all j, k  0. If {H

}

i0

and {K

}

i0

are

two bi-Hamiltonian hierarchies, then all functions

are in (bi-)involution provided that one of the two

hierarchies starts from a Casimir of { , }. In fact,

suppose that H

is such a Casimir. Then

; K

g¼fH

i1

; K

¼fH

i1

; K

jþ1

g¼

¼fH

; K

jþi

g¼0

and

; K

¼fH

iþ1

; K

g¼0

We observe that these proofs of the involutivity do

not use the compatibility condition [2] between the

Poisson structures. The point is that this condition is

important for the existence of bi-Hamiltonian hier-

archies. Indeed, the problem of the existence and the

construction of bi-Hamiltonian hierarchies is quite

delicate. We tackle it first in the case of a particular

class of bi-Hamiltonian manifolds, the so-called

Poisson–Nijenhuis manifolds. In turn, they are a

generalization of nondegenerate bi-Hamiltonian

manifolds.

Let (M, P, P

) be a bi-Hamiltonian manifold such

that P is invertible. Then we can introduce the

tensor field N = P

1

, which is of type (1, 1) and

will always be dealt with as an endomorphism of the

tangent bundle TM. This tensor field possesses some

remarkable properties. First of all, its Nijenhuis

torsion T(N) vanishes; this means that

TðNÞðX; YÞ¼½NX; NYN½X; Y

¼ 0

for any pair (X, Y) of vector fields on M, where

½X; Y

¼½NX; Yþ½X; NYN½X; Y

Multi-Hamiltonian Systems 463

Sometimes a tensor field with vanishing Nijenhuis

torsion is called a recursion operator. Since P defines

a symplectic structure on M, such a bi-Hamiltonian

manifold is called an !N manifold.

The tensor field N satisfies two compatibility

conditions with P. The first one is simply the

skew-symmetry of P

and reads NP = PN



, while

the second one is a restatement of [3],

½X

; X



¼ X

fF;Gg

8F; G 2 C

ðMÞ

A manifold is said to be a Poisson–Nijenhuis manifold

(briefly, a PN manifold) if it is endowed with a Poisson

tensor P and a torsionless (1, 1) tensor field N which

are compatible, in the sense that the two above-

mentioned conditions hold. We have just seen that

every nondegenerate bi-Hamiltonian manifold (i.e.,

such that one of the two Poisson tensors is invertible) is

a PN manifold. On the other hand, if (M, P, N)isa

PN manifold, then it can be shown that P

= NP is

a Poisson tensor, which is compatible with P.In

other words, PN manifolds are particular examples of

bi-Hamiltonian manifolds. Moreover, one has that

(j)

= N

P and P

(k)

= N

P are, for every j, k  0,

compatible Poisson tensors.

Let us consider now a function H

,onaPN

manifold (M, P, N), such that N



= dH

exact, where N



: T



M ! T



M is the adjoint of the

recursion operator N. This implies that

X ¼ P dH

¼ PN



¼ P

½17

is a bi-Hamiltonian vector field. By means of N



we can

define the 1-forms 

= (N



)

, which can be shown

to be all closed. If they are exact, that is, 

= dH

,then

the functions H

form a bi-Hamiltonian hierarchy and

thus are in involution. This shows that on a (simply

connected) PN manifold every bi-Hamiltonian vector

field of the form [17],withN



= dH

, belongs to a

bi-Hamiltonian hierarchy and that its first integrals (in

involution) can be iteratively constructed with the

recursion operator. (The integrability of this vector

field clearly depends on the number of independent

integrals of motion.) Moreover, the vector field

= P dH

k1

of the hierarchy is Hamiltonian

with respect to all Poisson structures P

(j)

with j  k,

because X

= P

(j)

kj

The example of the Toda lattice presented earlier

can be casted in the PN (more precisely, !N)

framework. One can introduce the recursion opera-

tor N and, in the three-particle case, one can define

the third integral of motion as dJ = N



dH. Since K,

H, and J belong to a bi-Hamiltonian hierarchy, they

are in involution, and this (along with their

functional independency) proves the integrability of

the Toda lattice.

In this example something more happens: the

integrals of motion are (up to multiplicative con-

stants) the traces of the powers of the recursion

operator N. This is a general fact, since the

vanishing of the torsion of N implies that N



kþ1

, where I

= (1=k)tr N

Next we deal with the case where the

bi-Hamiltonian manifold (M, P, P

) is not of the

Poisson–Nijenhuis type, that is, both P and P

are

degenerate. Let us suppose that their symplectic

leaves have codimension 1. We also want to discuss

in this case an iteration problem, namely the

problem of constructing a bi-Hamiltonian hierarchy

starting from a Casimir H

of P. Let us consider the

Hamiltonian vector field X

= P

= Y

(using

the notations introduced earlier). Thanks to the

form [4] of the compatibility condition between P

and P

, we have that

P ¼ L

P ¼L

¼ 0

meaning that X

is an infinitesimal symmetry of P.

Moreover, X

is tangent to the symplectic leaves of P,

since hdH

, X

i= hdH

, P

i= 0. Under some sui-

table topological assumptions, we can conclude that

there exists a function H

such that X

= P dH

,that

is, X

is a bi-Hamiltonian vector field. Now the

procedure can be iterated, that is, in the same way one

can show that, if X

= P

= Y

, then there exists

afunctionH

such that X

= P dH

, and so on. Thus,

one obtains a bi-Hamiltonian hierarchy {H

}

k0

which can either be infinite or end with a Casimir of

. In any case, the function H() =

k0



k

is a

Casimir of the Poisson pencil P



= P

 P.Asseen

earlier, the typical situation is that the chain terminates

with a Casimir H

of P

, where dim M = 2n þ 1. In

other words, there is a Casimir of the Poisson pencil

which is a polynomial of degree n in the parameter .

As a general procedure for constructing

bi-Hamiltonian hierarchies, one can look for the

Casimir functions H() of the Poisson pencil which

are deformations of Casimir functions of P, but it is

not clear when such a deformation does exist in the

case where the corank of the bi-Hamiltonian structure

is greater than 2. Nevertheless, suppose that H() =

k0



k

is a Casimir of P



, that is, that {H

}

k0

a bi-Hamiltonian hierarchy. Then, for all ,the

bi-Hamiltonian vector fields X

kþ1

= P dH

kþ1

= P

are Hamiltonian with respect to P



, with Hamiltonian

function H

(k)

() =

j = 0



kj

kþ1

¼ P



ðkÞ

ðÞ

Therefore, the vector fields X

are not only

bi-Hamiltonian, but they are Hamiltonian with

respect to any Poisson bracket of the pencil.

464 Multi-Hamiltonian Systems

In this article we have described some basic

properties of bi-Hamiltonian systems, defined on

manifolds possessing a Poisson pair. There are other

important vector fields on these manifolds (more

precisely, on !N manifolds). They are called cyclic

systems of Levi-Civita, and they give an intrinsic

description of the separable systems of Jacobi. We

refer to the article Recursion Operators in Classical

Mechanics in this encyclopedia for these topics.

See also: Bi-Hamiltonian Methods in Soliton Theory;

Classical r-Matrices, Lie Bialgebras, and Poisson Lie

Groups; Integrable Systems and Algebraic Geometry;

Integrable Systems and Recursion Operators on

Symplectic and Jacobi Manifolds; Integrable Systems:

Overview; Recursion Operators in Classical Mechanics;

Separation of Variables for Differential Equations;

Solitons and Kac–Moody Lie Algebras; Toda Lattices.

Further Reading

Adler M, van Moerbeke P, and Vanhaecke P (2004) Algebraic

Integrability, Painleve´ Geometry and Lie Algebras. Berlin:

Springer.

Arnol’d VI and Givental AB (2001) Symplectic geometry. In:

Arnol’d VI et al. (eds.) Encyclopedia of Mathematical

Sciences, Dynamical Systems IV, pp. 1–138. Berlin: Springer.

Babelon O, Cartier P, and Kosmann-Schwarzbach Y (eds.) (1994)

Lectures on Integrable Systems. Singapore: World Scientific.

Błaszak M (1998) Multi-Hamiltonian Theory of Dynamical

Systems. Berlin: Springer.

Dorfman I (1993) Dirac Structures and Integrability of Nonlinear

Evolution Equations. Chichester: Wiley.

Gelfand IM and Zakharevich I (1993) On the local geometry of a

bi-Hamiltonian structure. In: Corwin L et al. (eds.) The

Gelfand Mathematical Seminars 1990–1992, pp. 51–112.

Boston: Birka¨ user.

Magri F, Falqui G, and Pedroni M (2003) The method of Poisson

pairs in the theory of nonlinear PDEs. In: Conte R et al. (eds.)

Direct and Inverse Methods in Nonlinear Evolution Equations,

Lecture Notes in Physics, vol. 632, pp. 85–136. Berlin: Springer.

Magri F, Casati P, Falqui G, and Pedroni M (2004) Eight lectures

on integrable systems. In: Kosmann-Schwarzbach Y et al.

(eds.) Integrability of Nonlinear Systems, Lecture Notes in

Physics, vol. 638, pp. 209–250. Berlin: Springer.

Olshanetsky MA, Perelomov AM, Reyman AG, and Semenov-

Tian-Shansky MA (1994) Integrable systems. II. In: Arnol’d VI

et al. (eds.) Encyclopedia of Mathematical Sciences, Dynam-

ical systems VII, pp. 83–259. Berlin: Springer.

Olver PJ (1993) Applications of Lie Groups to Differential

Equations, 2nd edn. New York: Springer.

Vaisman I (1994) Lectures on the Geometry of Poisson Mani-

folds. Basel: Birkha¨user.

Multiscale Approaches

A Lesne, Universite

P.-M. Curie, Paris VI, Paris,

France

Introduction: Multiple-Scale

and Multiscale Approaches

Multiscale, or more precisely multiple-scale,

method is a technique of perturbation theory

based on the introduction of additional rescaled

variables, say time variables, formally considered as

independent variables and describing each a differ-

ent timescale (for the sake of simplicity, we will

mainly consider a dynamic framework and time-

scales; all can be transposed to spatial dependences

and scales). It was first developed to handle

singular situations in which dynamic regimes of

different characteristic scales coexist and intermin-

gle in such a way that straightforward perturbation

expansions are not uniformly convergent in time

(hence of limited relevance and use) due to the

so-called secular terms growing unbounded with

time; the freedom introduced together with the

extra variables indeed allows to impose conditions

preventing these secular divergences and improving

the convergence of the perturbation series. It yields

a global perturbation solution describing jointly the

behavior at small and large scales. This technique

belongs to the far more wide-ranging class of

multiscale approaches; these can be divided into

four main subclasses:

1. Mean-field techniques exploiting scale separation

between fast and slow components of the

dynamics. The influence of the slow variables

onto the fast dynamics, if any, is treated in a

decoupled way within a parametric approxima-

tion, allowing an adiabatic elimination of fast

variables (see the section ‘‘Slow/fast variables’’).

2. Singular pertur bations, in which individual fast

components ultimately give rise to slow trends

and influence the large-scale features. Scale

separation here breaks down at long times and

multiple-scale method is then a method of choice

(see the next section).

3. Matched expansions when regimes of different

scales succeed (boundary-layer singularity; see

the section ‘‘Boundary layers and matched

expansions’’).

Multiscale Approaches 465

4. Renormalization techniques, in systems exhibiting

some kind of universality in the relations between

their behaviors at different scales, for example,

scale invariance (see the section ‘‘Renormalization:

an iterated multiscale approach’’).

We will first present the principles of multiple-

scale method, detail its technical implementation on

simple abstract examples and cite some typical

applications. Then we will articulate this technique

with more general multiscale methods in a brief

overview (see the section ‘‘A brief overview of

multiscale approaches’’). The range of multiscale

approaches and technical tools will then be illus-

trated and compared in the context of diffusion,

Brownian motion, and transport phenomena (see the

section ‘‘Summary: the exemplary case of

diffusion’’).

Multiple-Scale Method: Principles

Context: Singular Perturbations and Secular

Divergences

Multiple-scale methods have been developed to

handle situations in which the dynamics involves a

small parameter  (e.g., the ratio of the masses of

different subsystems, the strength of an additional

interaction, the amplitude of an applied field)

directly controlling the separation between the

different characteristic timescales of the evolution

and, specifically, such that the behavior for  = 0 is

qualitatively different from the behavior for  small

(  1 but finite); in other words, when a weak

influence, of strength controlled by   1, does not

have only weak consequences. Typically, this occurs

when  represents the strength of a weak coupling

between otherwise independent subsystems or when

a vanishing value  = 0 changes a characteristic time,

the sign of a friction coefficient, the order of the

highest time derivative in case of ordinary differ-

ential equations (turning points), or the type of

partial differential equations in case of spatially

extended systems. Accordingly, a naive perturbative

approach with respect to , that is, an expansion

taking as a basic approximation the behavior for

 = 0, cannot bridge the qualitative gap with

behaviors observed for >0. It thus fails to give a

full account of the system evolution at all times: one

speaks of singular perturbation.

A historical example arose in celestial mechanics,

in the celebrated nonintegrable three-body problem,

involving the Sun, a big planet and a smaller one, of

respective masses m

, m

< m

and m

 m

. The

straightforward approach would be to consider the

presence of the small planet as a small perturbation

of the integrable two-body problem for the masses

and m

. But when one tries to determine the

solution as a series in powers of the mass ratio

 = m

, unbounded terms appear, the so-called

secular terms, increasing without bounds as fast as t,

hence of ill-defined order and impairing the very

consistency of the perturbation approach at long

times t > 1=. Accordingly, the perturbation expan-

sion is not uniformly convergent in time, preventing

from using it to investigate asymptotics and deter-

mine the fate of the three-body system: the influence

of the small planet on the motion of the bigger one,

although seemingly a weak perturbation, might

ultimately modify its trajectory around the Sun, at

least in some resonant cases.

The origin of secular terms lies in a phenomenon

of resonance, which is best explained on an

example: the Duffing oscillator

€

x þ x = x

with

  1. When looking for a solution in the form

x(t) =



(t), each component x

(t) has to be

bounded in order to get a consistent perturbation

expansion, in which the hierarchy of terms of

different orders remains valid forever: x

nþ1

(t) 

(t). These components should satisfy the following

sequence of equations:

€

þ x

¼ 0;

€

þ x

¼x

; ...

ðlinearized operator Lx 

€

x þ xÞ½1

It gives x

(t) = ae

þ c.c., from which follows a

secular contribution (3i=2)ajaj

t e

in x

(t). In

general, solving perturbatively

z = f (z, ) for an

expansion z(, t) =



(t) yields a hierarchical

sequence of equations of the form

= Lz

þ ’

, z

, ..., z

n1

) for n  1, where L = Df (z

,  = 0)

comes from the linearization in z

of the unperturbed

evolution law. A secular divergence arises in z

soon as ’

contains an additive contribution which is

an eigenvector of L (part of a mathematical result

known as the Fredholm alternative). The appearance

of secular terms reflects a singular feature of the

dynamics: the fact that the limits as  ! 0andt !1

do not commute. As a rule, such noninversion is

associated with generalized secular divergences: the

fast, short-term dynamics finally contributes to the

slow, long-term behavior. This feature is a clue

towards using multiple-scale method.

Technical Principles

The first step is to perform rescalings leading to

dimensionless variables and functions, which evidence

a small control parameter , related to scale separation

and providing a natural parameter for a perturbation

approach. The basic principle of multiple-scale

method is to introduce additional independent time

466 Multiscale Approaches