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

Подождите немного. Документ загружается.

Eliahou S, Kauffman LH, and Thistlethwaite MB (2003) Infinite

families of links with trivial Jones polynomial. Topology

42(1): 155–169.

Fateev VA and Zamolodchikov AB (1982) Self-dual solutions of

the star-triangle relations in Z

-models. Physics Letters A

92(1): 37–39.

Fredenhagen K, Rehren KH, and Schroer B (1989) Superselection

sectors with braid group statistics and exchange algebras.

Communications in Mathematical Physics 125: 201–226.

Freedman MH (2003) A magnetic model with a possible

Chern–Simons phase. With an appendix by F. Goodman

and H. Wenzl. C ommunications in Mathematic al Physics

234(1): 129–183.

Freyd P, Yetter D, Hoste J, Lickorish WBR, Millett K, and

Ocneanu A (1985) A new polynomial invariant of knots and

links. Bulletin of the American Mathematical Society (NS)

12(2): 239–246.

Haag R (1996) Local Quantum Physics. Berlin: Springer.

Jaeger F (1992) Strongly regular graphs and spin models for the

Kauffman polynomial. Geometriac Dedicata 44(1): 23–52.

Jimbo M (1986) A q-analogue of U(gl(N þ 1)), Hecke algebra,

and the Yang–Baxter equation. Letters in Mathematical

Physics 11(3): 247–252.

Jones VFR (1983) Index for subfactors. Inventiones Mathemati-

cae 72: 1–25.

Jones VFR (1985) A polynomial invariant for knots via von

Neumann algebras. Bulletin of the American Mathematical

Society 12: 103–112.

Jones VFR (1987) Hecke algebra representations of braid groups

and link polynomials. Annals of Mathematics 126(2):

335–388.

Jones V (1989) On knot invariants related to some statistical

mechanical models. Pacific Journal of Mathematics 137:

311–388.

Kauffman LH (1987) State models and the Jones polynomial.

Topology 26(3): 395–407.

Kauffman LH (1990) An invariant of regular isotopy. Transac-

tions of the American Mathematical Society 318(2): 417–471.

Khovanov M (2000) A categorification of the Jones polynomial.

Duke Mathematical Journal 101(3): 359–426.

Lickorish WBR (1997) An Introduction to Knot Theory, Graduate

Texts in Mathematics, vol. 175. New York: Springer.

Long DD and Paton M (1993) The Burau representation is not

faithful for n  6. Topology 32(2): 439–447.

Longo R (1989) Index of subfactors and statistics of quantum

fields I. Communications in Mathematical Physics 126:

217–247.

Longo R (1990) Index of sub factor and statistics of quantum

fields II. Communications in Mathematical Physics 130:

285–309.

Moody JA (1991) The Burau representation of the braid group B

is unfaithful for large n. Bulletin of the American Mathematical

Society (NS) 25(2): 379–384.

Murakami J (1990) The representations of the q-analogue of

Brauer’s centralizer algebras and the Kauffman polynomial of

links. Publ. Res. Inst. Math. Sci. 26(6): 935–945.

Murasugi K (1987) Jones polynomials and classical conjectures in

knot theory. Topology 26(2): 187–194.

Pimsner M and Popa S (1986) Entropy and index for subfactors.

Annales Scientifiques de l’Ecole Normale Supe´rieure 19: 57–106.

Przytycki JH and Traczyk P (1988) Invariants of links of Conway

type. Kobe Journal of Mathematics 4(2): 115–139.

Rosso M (1988) Groupes quantiques et mode` les vertex de

V. Jones en the´orie des nœuds. (French) [Quantum groups

and V. Jones’s vertex models for knots]. Comptes Rendus de

l’Academie des Sciences Paris Se´ries I Mathematiques 307(6):

207–210.

Reshetikhin N and Turaev VG (1991) Invariants of 3-manifolds

via link polynomials and quantum groups. Inventiones

Mathematicae 103(3): 547–597.

Temperley HNV and Lieb EH (1971) Relations between the

‘percolation’ and ‘colouring’ problem and other graph-

theoretical problems associated with regular planar lattices:

some exact results for the ‘percolation’ problem. Proceedings

of the Royal Society of London Series A 322(1549): 251–280.

Thistlethwaite MB (1987) A spanning tree expansion of the Jones

polynomial. Topology 26(3): 297–309.

Thistlethwaite M (2001) Links with trivial Jones polynomial.

Journal of Knot Theory and Its Ramifications 10(4): 641–643.

Tsuchiya A and Kanie Y (1988) Vertex operators in conformal

field theory on P

and monodromy representations of braid

group. In: Conformal Field Theory and Solvable Lattice

Models, (Kyoto, 1986), Adv. Stud. Pure Math., vol. 16,

pp. 297–372. Boston, MA: Academic Press.

Turaev VG (1994) Quantum Invariants of Knots and 3-Manifolds.

de Gruyter Studies in Mathematics, vol. 18. Berlin: Walter de

Gruyter.

Vassiliev VA (1990) Cohomology of knot spaces. In: Theory of

Singularities and Its Applications, Advances in Soviet Mathe-

matics, vol. 1, pp. 23–69. Providence, RI: American Mathema-

tical Society.

Wassermann A (1998) Operator algebras and conformal field

theory. III. Fusion of positive energy representations of

LSU(N) using bounded operators. Inventiones Mathematical

133(3): 467–538.

Wenzl H (1987) On sequences of projections. C. R. Math. Rep.

Acad. Sci. Canada 9(1): 5–9.

Wenzl H (1988) Hecke algebras of type A

and subfactors.

Inventiones Mathematical 92(2): 349–383.

Wenzl H (1990) Quantum groups and subfactors of type B, C,

and D. Communications in Mathematical Physics 133(2):

383–432.

Witten E (1988) Topological quantum field theory. Communica-

tions in Mathematical Physics 117(3): 353–386.

Witten E (1989) Quantum field theory and the Jones polynomial.

Communication in Mathematical Physics 121(3): 351–399.

The Jones Polynomial 187

Kac–Moody Lie Algebras see Solitons and Kac–Moody Lie Algebras

KAM Theory and Celestial Mechanics

L Chierchia, Universita

degli Studi ‘‘Roma Tre,’’

Rome, Italy

Introduction

Kolmogorov–Arnol’d–Moser (KAM) theory deals

with the construction of quasiperiodic trajectories

in nearly integrable Hamiltonian systems and it was

motivated by classical problems in celestial

mechanics such as the n-body problem. Notwith-

standing the formidable bulk of results, ideas and

techniques produced by the founders of the modern

theory of dynamical systems, most notably by

H Poincare´ and G D Birkhoff, the fundamental

question about the persistence under small perturba-

tions of invariant tori of an integrable Hamiltonian

system remained completely open until 1954. In that

year, A N Kolmogorov stated what is now usually

referred to as the KAM theorem (in the real-analytic

setting) and gave a precise outline of its proof,

presenting a strikingly new and powerful method to

overcome the so-called small-divisor problem (reso-

nances in Hamiltonian dynamics produce, in the

perturbation series, divisors which may become

arbitrarily small, making convergence argument

extremely delicate). Subsequent ly, KAM theory has

been extended and applied to a large variety of

different problems, including infinite-dimensi onal

dynamical systems and partial differential equations

with Hamiltonian structure. However, establishing

the existence of quasiperiodic motions in the n-body

problem turned out to be a longer story, which only

very recently has reached a satisfactory level; the

point being that the n-body problems present strong

degeneracies, which violate the main hypotheses of

the KAM theorem.

This article gives an account of the ideas and

results concerning the construction of quasiperiodic

solutions in the planetary n-body problem. The

synopsis of the article is the following.

The next section gives the analytical description of

the planet ary (1 þ n)-body proble m.

In the sub section ‘‘Kolmogo rov’s theore m and the

RPC3 BP (1954 ),’’ original version of the KAM

theorem is recalled, giving an outline of its proof

and showing its implications for the simplest many-

body case, namely, the restri cted, planar, and

circular three-body problem.

In the section ‘‘Arn ol’d’s theorem, ’’ the existenc e

of a posit ive measure set of initial data in phase

space giving rise to quasiperiodic motions near

coplanar and nearly circular unperturbed Keplerian

trajectories is presented. The rest of the section is

devoted to the proof of Arnol’d’s theorem following

the historical developments: Arnol’d’s proof (1963a)

for the planar three-body case is presented, the

extension to the spatial three-body case due to

Laskar and Robutel (1995) is discussed, and Her-

man’s proof – in the form given by Fe´joz in 2004 –

of the genera l spatial (1 þ n)-case is prese nted.

In the sect ion ‘‘Low er dimensio nal tori,’’ a brief

discussion of the construction of lower-dimensional

elliptic tori bifurcating from the Keplerian unper-

turbed motions is given (these results have been

established in the early 2000s).

Finally, the problem of taking into account real

astronomical parameter values is considered and a

recent result on an application of (computer-

assisted) KAM techniques to the solar subsystem

formed by Sun, Jupiter, and the asteroid Victoria is

briefly mentioned.

The Planetary (1 þ n)-Body Problem

The evolution of (1 þ n)-body systems (assimilated

to point masses) interacting only through gravita-

tional attraction is governe d by Newton’s equations.

If u

(i)

2 R

denotes the position of the ith body in a

given reference frame and if m

denotes its mass,

then Newton’s equations read

ðiÞ

¼

0jn

j6¼i

ðiÞ

 u

ðjÞ

ðiÞ

 u

ðjÞ

; i ¼ 0; 1; ...; n ½1

Here the gravitational constant is taken to be equal

to 1 ( which amounts to rescale the time t).

Equations [1] are equivalent to the standard

Hamilton’s equations corresponding to the Hamil-

tonian function

New

:¼

i¼0

ðiÞ



0i<jn

ðiÞ

 u

ðjÞ

½2

where (U

(i)

, u

(i)

) are standard symplectic variables

and the phase space is the ‘‘collisionless domain’’

M:= {U

(i)

, u

(i)

2 R

: u

(i)

6¼ u

(j)

,0 i 6¼ j  n}; the

symplectic form is the standard one:

(i)

i, k

(i)

^ du

(i)

; jj denotes the standard

Euclidean norm. Introducing the symplectic coordi-

nate change (U, u) = 

hel

(R, r),



hel

ð0Þ

¼r

ð0Þ

; u

ðiÞ

¼r

ð0Þ

þr

ðiÞ

ði ¼ 1; ...; nÞ

ð0Þ

¼R

ð0Þ



i¼1

ðiÞ

; U

ðiÞ

¼R

ðiÞ

ði ¼ 1; ...; nÞ

½3

one sees that the Hamiltonian H

hel

:= H

New



hel

does not depend upon r

(0)

(recall that a local

diffeomorphism is called symplec tic if it preserves

the symplec tic form). This means that R

(0)

( total

linear momentum) is a global integral of motion.

Without loss of generality, one can restrict attention

to the invariant manifold M

:= {R

(0)

= 0} (invar-

iance of eqn [1] by changes of inertial reference

frames).

In the ‘‘planetary’’ case, one assumes that one of

the bodies, say i = 0 (the Sun), has mass much larger

than that of the other bodies (this accounts for the

index ‘‘hel,’’ which stands for ‘‘heliocentric’’). To

make the perturbative character of the problem

transparent, one may intr oduce the following rescal-

ings. Let

¼ "



; X

ðiÞ

5=3

; x

ðiÞ

2=3

ði ¼1; ...; nÞ½4

and rescale time by a factor "m

7=3

(which amount s

to dividing the new Hamiltonian by such a

factor); then, the flow of the Hamiltonian H

hel

is equivalent to the flow of the Hamiltonian

plt

:¼

i¼1

ðiÞ

2





ðiÞ

þ "

1i<jn

ðiÞ

 X

ðjÞ



ðiÞ

 x

ðjÞ



½5

on the phase space M:= {X

(i)

, x

(i)

2 R

:1 i  n

and 0 6¼ x

(i)

6¼ x

(j)

} with respect to the standard

symplectic form

i = 1

(i)

^ dx

(i)

; the mass para-

meters are defined as

:¼ 1 þ "



;

:¼



þ "



½6

The following observations can be made:

1. The Hamiltonian

ð0Þ

plt

:¼

i¼1

ðiÞ

2





ðiÞ

is integrable and represents the sum of n two-

body systems formed by the Sun and the ith

planet (disregarding the interaction with the

other planets).

2. The transformation 

hel

in eqn [3] preserves

the total angular momentum

C :=

i = 0

(i)



(i)

, which is a vector-valued integral for

New

. Thus, the three components, C

,of

C :=

i = 1

(i)

 x

(i)

(which is proportional to

C and is termed the ‘‘total angular momen-

tum’’), are integrals for H

plt

.TheintegralsC

do not commute: if {,} denotes the standard

Poisson bracket, then {C

, C

} = C

(and, cycli-

cally, {C

, C

} = C

,{C

} = C

). Neve rtheless,

one can form two (independent) commuting

integrals, for example, jCj

and C

.Thisshows

that the (spatial) (1 þ n)-body problem has

(3n  2) degrees of freedom.

3. An important special case is the planar (1 þ n)-

body problem. In such a case, one assumes that

all the ‘‘single’’ angular momenta C

(i)

:= X

(i)

 x

(i)

are parallel. In this case, the motion takes place

on a fixed plane orthogonal to C and (up to a

rotation of the reference frame) one can take, as

symplectic variables, X

(i)

, x

(i)

2 R

. The Hamilto-

nian H

pln

governing the dynamics of the planar

(1 þ n)-body problem is, then, given on the right-

hand side of eqn [5] with X

(i)

, x

(i)

2 R

. Notice

that the planar (1 þ n)-body problem has 2n

degrees of freedom.

4. For a deeper understanding of the perturbation

theory of the planetary many-body problem, it is

necessary to find ‘‘good’’ sets of symplectic

coordinates, which the founders of celestial

190 KAM Theory and Celestial Mechanics

mechanics (most notably, Jacobi, Delaunay, and

Poincare´) have done. In particular, Delaunay

introduced an analytic set of symplectic ‘‘action-

angle’’ variables. Recall the Delaunay variables

for the two-body ‘‘reduced Hamiltonian’’

Kep

jXj

2



M

jxj

Let {k

, k

} be a standard orthonormal basis

in the x-configuration space; let the angular

momentum C = X  x be nonparallel to k

and

let the energy E = H

Kep

< 0. In such a case, x(t)

describes an ellipse lying in the plane orthogonal

to C, with focus in the origin and fixed symmetry

axes. Let a be the semimajor axis of the ellipse

spanned by x; { (the inclination) be the angle

between k

and C; G = jCj; =G cos { = C  k

;

L = m

ﬃﬃﬃﬃﬃﬃﬃ

; ‘ be the mean anomaly of x (:= 2

times the normalized area spanned by x mea-

sured from the perihelion P, which is the point

of the ellipse closest to the origin);  be the

angle between k

and N := k

 C (:= oriented

‘‘node’’); and g be the argument of the perihelion

(:= the angle between N and (O, P)). Then

(letting T := R=(2Z))

ðL; G; Þ2fL > 0gfG >  > 0g

ð‘; g;Þ2T

½7

are conjugate symplectic coordinates and if 

Del

is the corresponding symplectic map, then

Kep

 

Del

= (

)=(2L

Note that the Delaunay variables become

singular when C is vertical (the node is no more

defined) and in the circular limit (the perihelion

is not unique). In these cases different variables

have to be used.

5. Let (X

(i)

, x

(i)

) = 

Del

((L

, G

, 

), (‘

, g

, 

)). Then

plt

expressed in the Delaunay variables

{(L

, G

, 

), (‘

, g

, 

): 1  i  n} becomes

Del

¼H

ð0Þ

Del

þ"H

ð1Þ

Del

; H

ð0Þ

Del

:¼

i¼1



½8

Note that the number of action variables on

which the integrable Hamiltonian H

(0)

Del

depends

is strictly less than the number of degrees of

freedom. This ‘‘proper degeneracy,’’ as we shall

see in nex t sections, brings in an essential

difficulty one has to face in the perturbative

approach to the many-body problem. In fact, this

feature of the many-body problem is common to

several other problems of celestial mechanics.

Maximal KAM Tori

Kolmogorov’s Theorem and the RPC3BP (1954)

Kolmogorov’s invariant tori theorem deals with

the persistence, in nearly integrable Hamiltonian

systems, of Lagrangian (maximal) tori, which, in

general, foliate the integrable lim it. Kolmogorov

(1954) stated his theorem and gave a precise

outline of the proof. Let us briefly recall this

milestone of the modern theory of dynamical

systems.

Let M:= B

 T

being a d-dimensional ball

in R

centered at the origin) be endowed with the

standard symplectic form dy ^ dx :=

^ dx

(y 2 B

, x 2 T

). A Hamiltonian function N on M

having a Lagrangian invariant d-torus of energy E

on which the N-flow is conjugated to the linear

dense translation x ! x þ !t, ! 2 R

can be

put in the form

N :¼ E þ !  y þ Qðy; xÞ



Qð0; xÞ¼0; 8 2 N

; jj1

½9

(as usual, jj= 

þþ

, !y :=

i = 1

and @



= @



@



); in such a case, the Hamiltonian

N is said to be in Kolmogorov normal form. The

vector ! is called the ‘‘frequency vector’’ of the

invariant torus {y = 0}  T

. The Hamiltonian N is

said to be nondegenerate if

deth@

Qð0; Þi 6¼ 0 ½10

where the brackets denote average over T

and @

the Hessian with respect to the y-variables.

We recall that a vector ! 2 R

is said to be

‘‘Diophantine’’ if there exist >0 and   d  1

such that

j!  kj



jkj



; 8k 2 Z

nf0g½11

The set D

of all Diophantine vectors in R

is a set of

full Lebesgue measure. We also recall that Hamilto-

nian trajectory is called quasiperiodic with (rationally

independent) frequency ! 2 R

if it is conjugate to

the linear translation  2 T

! þ !t 2 T

Theorem (Kolmogorov 1954) Consider a one-

parameter family of real-analytic Hamiltonian func-

tions H

:= N þ "P where N is in Kolmogorov normal

form (as in eqn [9]) and " 2 R. Assume that ! is

Diophantine and that N is nondegenerate. Then,

there exists "

> 0 and for any j"j"

, areal-analytic

symplectic transformation 

: M!M putting H

Kolmogorov normal form, H

 

= N

, with

:= E

þ !  y

þ Q

, x

). Furthermore, jE

 Ej,

k

 idk

, and kQ

 Qk

are small with ".

KAM Theory and Celestial Mechanics 191

In other words, the Lagrangian unpertur bed torus

:= {y = 0}  T

persists under small perturbation

and is smoothly deformed into the H

-invariant

torus T

:= 

({y

= 0}  T

); the dynamics on such

torus, for all j"j"

, consists of dense quasiperiodic

trajectories. Note that the H

-flow on T

is analytically conjugated by 

to the translation

þ !t with the same frequency vector of N,

while the energy of T

, namely E

, is in general

different from the energy E of T

Kolmogorov’s proof is based on an iterative

(Newton) scheme. The map 

is obtained

as lim

k !1



(1)



(k)

, where the 

(j)

’s are

("-dependent) symplectic transformations of M

successively closer to the identity. It is enough

to describe the construction of 

(1)

; 

(2)

then obtained by replacing H

with H

 

(1)

andsoon.Themap

(1)

is "-c lose to the identity

and it is generated by g(y

, x):= y

x þ

"(bx þ s(x) þ y

a(x)), where s and a are (res p.

scalar- and vector-valued) real-analytic functions

on T

with z ero av erage and b 2 R

; this means

that the symplect ic map 

(1)

:(y

, x

) !(y, x)is

implicitly given by the relations y = @

g and

= @

g. It is easy to see that the re e xists a unique

g of the above form such that for a suitable "

> 0,

 

ð1Þ

¼ E

þ !  y

þ Q

ðy

; x

Þþ"

8j"j"

½12

with @



(0, x

) = 0, for any  2 N

and jj1; here,

, Q

,andP

depend on " and, for a suitable c

> 0

and for j"j"

, jE  E

jc

j"j, kQ  Q

 c

j"j,

and kP

 c

Notice that the symplectic transformation 

(1)

actually the composition of two ‘ ‘elementary’’ transfo-

mations: 

(1)

= 

(1)

 

(1)

where 

(1)

:(y

, x

) !(, )

is the symplectic lift of the T

-diffeomorphism given

by x =  þ "a() (i.e., 

(1)

is the symplectic map

generated by y

  þ "y

 a()), while 

(1)

:(, ) !

(y, x) is the angle-dependent action translation gener-

ated by   x þ "(b  x þ s(x)); 

(1)

acts in the ‘ ‘angle

direction’’ and straightens out the flow up to order

O("

), while 

(1)

acts in the ‘ ‘action direction’’ and is

needed to keep the frequency of the torus fixed.

Since H

 

=: N

þ "

is again a perturbation

of a nondegenerate Kolmogorov normal form (with

same frequency vector !), one can repeat the

construction by obtaining a new Hamiltonian of

the form N

þ "

. Iterating, after k steps, one gets

a Hamiltonian N

þ "

. Carrying out the

(straightforward but lengthy) estimates, one can

check that kP

 c

, for a suitable constant

c > 1 independent of k (the fast growth of the

constant c

is due to the presence of the small

divisors appearing in the explicit construction of the

symplectic transformations 

(j)

). Thus , it is clear that

taking "

small enough the iterative procedure

converges (superexponentially fast) yielding the

thesis of the above theorem.

6. While the statement of the invariant tori theorem

and the outline of the proof are very clearly

explained in Kolmogorov (1954), Kolmogorov

did not fill out the details nor gave any estimates.

Some years later, Arnol’d (1963a) published a

detailed proof, which, however, did not follow

Kolmogorov’s idea. In the same year, J K Moser

published his invariant curve theorem (for area-

preserving twist diffeomorphisms of the annulus)

in smooth setting. The bulk of techniques and

theorems stemmed out from these works is

normally referred to as KAM theory; for reviews,

see Arnol’d (1988) or Bost (1984–85). A very

complete version of the ‘‘KAM theorem’’ both in

the real-analytic and in the smooth case (with

optimal smoothness assumptions) is given in

Salamon (2004); the proof of the real-analytic

part is based on Kolmogorov’s scheme. The

KAM theory of M Herman, used in his approach

to the planetary problem, is based on the abstract

functional theoretical approach of R Hamilton

(which, in turn, is a development of Nash–Moser

implicit function theorem; see Bost (1984–85) for

references); it is interesting, however, to note that

the heart of Herman’s KAM method is based on

the above-mentioned Kolmogorov’s transforma-

tion 

(1)

(compare Fe´joz (2002)).

7. In the nearly integrable case, one considers a one-

parameter family of Hamiltonians H

(I) þ "H

(I, x)

with (I, x) 2M:= U  T

standard symplectic

action-angle variables, U being an open subset of

.When" = 0, the phase space M is foliated

by H

-invariant tori {I

}  T

,onwhichtheflow

is given by x ! x þ @

)t.IfI

such that ! := @

) is Diophantine and if

det @

) 6¼ 0, then from Kolmogorov’s theorem

it follows that the torus {I

}  T

persists under

perturbation. In fact, introduce the symplectic

variables (y, x)withy = I  I

and let N(y):=

þ y), which by Taylor’s formula can be

written as H

) þ !  y þ Q(y)withQ(y)quad-

ratic in y and @

Q(0) = @

) invertible. One can

then apply Kolmogorov’s theorem with P

(y, x):=

þ y, x).

Notice that Kolmogorov’s nondegeneracy con-

dition det @

) 6¼ 0 simply means that the

frequency map

I 2 B

 U ! !ðIÞ :¼ @

ðIÞ½13

192 KAM Theory and Celestial Mechanics

is a local diffeomorphism (B

being a ball

around I

8. The symplectic structure imp lies that if n denotes

the number of degrees of freedom (i.e., half of the

dimension of the phase space) and d is the

number of independent frequencies of a quasi-

periodic motion, then d  n;ifd = n, the quasi-

periodic motion is called maximal. Kolmogorov’s

theorem gives sufficient conditions in order to get

maximal quasiperiodic solutions. In fact, Kolmo-

gorov’s nondegeneracy condition is an open

condition and the set of Diophantine vectors is

a set of full Lebesgue measure. Thus, in general,

Kolmogorov’s theorem yields a positive invariant

measure set spann ed by maximal quasiperiodic

trajectories.

As mentioned above, the planetary many-body

models are properly degenerate and violate

Kolmogorov’s nondegeneracy conditions and,

hence, Kolmogorov’s theorem – clearly motivated

by celestial mechanics – cannot be applied.

There is, however, an important case to which a

slight variation of Kolmogorov’s theorem can be

applied (Kolmogorov did not mention this in 1954).

The case referred to here is the simplest nontrivial

three-body problem, namely, the restricted, planar,

and circular three-body problem (RPC3BP for short).

This model, largely investigated by Poincare´, deals

with an asteroid of ‘‘zero mass’’ moving on the plane

containing the trajectory of two unperturbed major

bodies (say, Sun and Jupiter) revolving on a Keplerian

circle. The mathematical model for the restricted

three-body problem is obtained by taking n = 2and

setting m

= 0ineqn [1]: the equations for the two

major bodies (i = 0, 1) decouple from the equation

for the asteroid (i = 2) and form an integrable two-

body system; the problem then consists in studying

the evolution of the asteroid u

(2)

(t) in the given

gravitational field of the primaries. In the circular

and planar cases, the motion of the two primaries is

assumed to be circular and the motion of the

asteroid is assumed to take place on the plane

containing the motion of the two primaries; in fact

(to avoid collisions), one considers either inner or

outer (with respect to the circle described by the

relative motion of the primaries) asteroid motions.

To descr ibe the Hamiltonian H

rcp

governing the

motion of the RCP3BP problem, introduce planar

Delaunay variables ((L, G), (‘,

g)) for the asteroid

(better, for the reduced heliocentric Sun–asteroid

system). Such variables, which are closely related to

the above (spatial) Delaunay variables, have the

following physical interpretation: G is proportional

to the absolute value of the angular momentum of

the asteroid, L is proportional to the square root of

the semimajor axis of the instantaneous Sun–

asteroid ellipse, ‘ is the mean anomaly of the

asteroid, while

g the argument of the perihelion.

Then, in suitably normalized units, the Hamiltonian

governing the RPC3BP is given by

rcp

ðL; G;‘;g; "Þ:¼

 G

þ "H

ðL; G;‘;g; "Þ½14

where g :=

g  ,  2 T being the longitude of Jupi-

ter; the variables ((L, G), (‘, g)) are symplectic coordi-

nates (with respect to the standard symplectic form);

the normalizations have been chosen so that the

relative motion of the primary bodies is 2 periodic

and their distance is 1; the parameter " is (essentially)

the ratio between the masses of the primaries; the

perturbation H

is the function x

(2)

x

(1)

 1=jx

(2)



(1)

j expressed in the above variables, x

(2)

being the

heliocentric coordinate of the asteroid and x

(1)

that of

the planet (Jupiter): such a function is real-analytic on

{0 < G < L}  T

and for small " (for complete

details, see, e.g., Celletti and Chierchia (2003)).

The integrable limit

ð0Þ

rcp

:=H

rcp

" = 0

= 1=ð2L

ÞG

has vanishing Hessian and, hence, violates

Kolmogorov’s nondegeneracy condition (as

described in item (7) above). However, there is

another nondegeneracy condition which leads to a

simple variation of Kolmogorov’s theorem, as

explained briefly below.

Kolmogorov’s nondegeneracy condition det

) 6¼ 0 allows one to fix d-parameters, namely, the

d-components of the (Diophantine) frequency vector

! = @

). Instead of fixing such parameters, one

may fix the energy E = H

) together with the

direction {s! : s 2 R} of the frequency vector: for

example, in a neighborhood where !

6¼ 0, one can

fix E and !

for 1  i  d  1. Notice also that if

! is Diophantine, then so is s! for any s 6¼ 0 (with

same  and rescaled ). Now, it is easy to check that

the map I 2 H

1

(E) ! (!

, ..., !

d1

) is (at

fixed energy E) a local diffeomorphism if and only if

the (d þ 1)  (d þ 1) matrix

evaluated at I

is invertible (here the vector @

the upper right corner has to be interpreted as a

column while the vector @

in the lower left

corner has to be interpreted as a row). Such

KAM Theory and Celestial Mechanics 193

‘‘iso-energetic nondegeneracy’’ condition, rephrased

in terms of Kolmogorov’s normal forms, becomes

det

Qð0; Þi !

! 0



6¼0 ½15

Kolmogorov’s theorem can be easily adapted to the

fixed energy case. Assuming that ! is Diophantine

and that N is isoenergetically nondegenerate, the

same conclusion as in Kolmogorov’s theorem holds

with N

:= E þ !

 y

þ Q

, x

), where !

= 

and j

 1j is small with ".

In the RCP3BP case, the isoenergetic nondegene-

racy is met, since

det

ðL;GÞ

ð0Þ

rcp

ðL;GÞ

ð0Þ

rcp

ðL;GÞ

ð0Þ

rcp

Therefore, one can conclude that on each negative

energy level, the RCP3BP admits a positive measure

set of phase points, whose time evolution lies on two-

dimensional invariant tori (on which the flow is

analytically conjugate to linear translation by a

Diophantine vector), provided the mass ratio of the

primary bodies is small enough; such persistent tori

are a slight deformation of the unperturbed ‘‘Kepler-

ian’’ tori corresponding to the asteroid and the Sun

revolving on a Keplerian ellipse on the plane where

the Sun and the major planet describe a circular orbit.

In fact, one can say more. The phase space for the

RCP3BP is four dimensional, the energy levels are

three dimensional, and Kolmogorov’s invariant tori

are two dimensional. Thus, a Kolmogorov torus

separates the energy level, on which it lies, into two

invariant components, and two Kolmogorov’s tori

form the boundary of a compact invariant region so

that any motion starting insuchregionwillnever

leave it. Thus, the RCP3BP is ‘‘totally stable’’: in a

neighborhood of any phase point of negative energy, if

the mass ratio of the primary bodies is small enough,

the asteroid stays forever on a nearly Keplerian ellipse

with nearly fixed orbital elements L and G.

Arnol’d’s Theorem

Consider again the planetary (1 þ n)-body problem

governed by the Hamiltonian H

plt

in eqn [5]. In the

integrable approximation, governed by the Hamil-

tonian H

(0)

plt

, the n planets describe Kepler ian ellipses

focused on the Sun. Arnol’d (1963b) has stated the

following theorem.

Theorem (Arnol’d 1963b) Let ">0 be small

enough. Then, there exists a bounded, H

plt

-invariant

set F(") Mof positive Lebesgue measure corre-

sponding to planetary motions with bounded

relative distances; F(0) corresponds to Keplerian

ellipses with small eccentricities and small relative

inclinations.

This theorem represents a major achievement in

celestial mechanics solving more than tri-c

entennial

mathematical problem. Arnol’d (1963b) gave a

complete proof of this result only in the planar

three-body case and gave some indications of how to

extend his approach to the general situation.

However, to give a full proof of Arnol’d’s theorem

in the general case turned out to be more than a

technical problem and new ideas were needed: the

complete proof (due, essentially, to M Herman) has

been given only in 2004.

In the following subsections, we briefly review

the history and the ideas related to the proof of

Arnol’d’s theorem. As for credits: the proof of Arnol’d’s

theorem in the planar 3BP case is due to Arnol’d himself

(Arn ol’d 1963b); the spatial 3BP case is due to Laskar

and Robutel (1995) and Robutel (1995); the general

case is due to Herman (1998) and Fe´joz (2004).The

exposition we have given does not always follow the

original references.

The planar three-body problem Recall the Hamil-

tonian H

pln

of the planar (1 þ n )-body pro blem

given in item (3) of the sect ion ‘‘The planetary

(1 þ n)-body proble m.’’ A co nvenient set of sym-

plectic variables for nearly circular motions are the

‘‘planar Poincare´ variables.’’ To describe such vari-

ables, consider a single, planar two-body system

with Hamiltonian

jXj

2



M

jxj

; X 2 R

; 0 6¼ x 2 R

ðwith respect to dX ^ dxÞ

½16

and introduce – as done before formula [14] for

(0)

rcp

– planar Delaunay variables ((L, G), (‘, g))

(here, g =

g = argument of the perihelion). To remove

the singularity of the Delaunay variables near zero

eccentricities, Poincare´introducedvariables

((, ), (, )) defined by the following formulas:

 ¼ L; H ¼ L  G

 ¼ ‘ þ g; h ¼g

ﬃﬃﬃﬃﬃﬃﬃ

cos h ¼ 

ﬃﬃﬃﬃﬃﬃﬃ

sin h ¼ 

½17

As Poincare´ showed, such variables are symplectic and

analytic in a neighborho od of (0, 1)  T  {0, 0};

notice that the symplectic map ((, ), (, )) !(X, x)

depends on the parameters , M,and". In Poincare´

variables, the two-body Hamiltonian in eqn [16]

194 KAM Theory and Celestial Mechanics

becomes =(2

), with  := (=m

)

=M.Now,

re-insert the index i,let

:((

, 

), (

, 

)) ! (X

(i)

)and(, , , ) = (

(

, 

, 

, 

), ..., 

(



, 

, 

)). Then, the Hamiltonian for the planar

(1 þ n)-body problem takes the form

pln

  ¼H

ðÞþ"H

ð;;;Þ

:¼

i¼1





;

:¼





:¼H

compl

þH

princ

½18

where the so-called ‘‘complementary part’’ H

compl

and the ‘‘principal part’’ H

princ

of the perturbation

are, respectively, the functions

1i<jn

ðiÞ

X

ðjÞ

and

1i<jn



ðiÞ

x

ðjÞ

½19

expressed in Poincare´ variables.

The scheme of proof of Arnol’d’s theorem in the

planar, three-body case (one star, n = 2 planets) is as

follows. The Hamiltonian is given by eqn [13] with

n = 2; the phase space is eight dimensional (four

degrees of freedom). This system, as mentioned several

times, is properly degenerate and Kolmogorov’s

theorem cannot be applied directly; furthermore, a

full (four-dimens ional) set of action variables needs

to be identified.

A first observation is that, in the planetary model,

there are ‘‘fast variables’’ (the 

’s describing the

revolutions of the planets) and ‘‘secular variables’’

(the 

’s and 

’s describing the variations of position

and shape of the instantaneuous Keplerian ellipses).

By averaging theory (see, e.g., Arnol’d (1998)), one

can ‘‘neglect,’’ in nonresonant regions, the fast-angle

dependence up to high order in " obtaining an

effective Hamiltonian, which, up to O("

), is given

by the ‘‘secular’’ Hamiltonian

sec

:¼H

ðÞþ"



ð;;Þ



ð;;Þ:¼

d

ð2Þ

½20

‘‘Nonresonant region’’ means, here, an open -set

where @



 k 6¼ 0 for k 2 Z

, jk

jþjk

jK and

for a suitable K  1.

In order to analyze the secular Hamiltonian, we

shall beriefly consider



as a function of the

symplectic variables  and , regarding the ‘‘slow

actions’’ 

as parameters.

For symmetry reasons,



is even in (, ) and the

point (, ) = (0, 0) is an elliptic equilibrium for



the eigenvalues of the matrix S@

(, )



(,0,0),

S being the standard symplectic matrix, are purely

imaginary numbers { i

, i

}. The real numbers

{

} are symplectic invariants of the secula r Hamil-

tonian and are usually called first (or linear) Birkh-

off invariants. In a neighborhood of an elliptic

equilibrium, one can use Birkhoff’s normal form

theory (see, e.g., Siegel (1971)): if the linear

invariants (

, 

) are nonresonant up to order r

(i.e., if   k :=

þ 

6¼ 0 for any k 2 Z

such that jk

jþjk

jr), then one can find a

symplectic transformation 

Bir

so that



 

Bir

¼ FðJ

; J

;Þþo

; J



þ 

½21

where F is a polynom ial of degree [r=2] of the form



þ 

þ (1=2)MJ  J þ, M= M() being a

(2  2) matrix (and o

=jJj

r=2

!0asjJj!0). Arnol’d,

using computations performed by Le Verrier,

checked the nonresonance condition up to order

r = 6 in the asymptotic regime a

!0 (where a

denote the sem imajor axes of approximate Kepler-

ian ellipses of the two planets); these computations

represent one of the most delicate parts of the paper.

Thus, combining averaging theory and Birkhoff

normal form t heory, one can construct a symplec-

tic change of variables defined on an open

subset of the phase space (avoiding some linear

resonances) (, , , ) !(

, 

, J, ’), where 

i

ﬃﬃﬃﬃﬃﬃ

exp (i’

), casting the three-body Hamil-

tonian into the form

ð

Þþ" ð

ÞJ þ

Mð

ÞJ  J



þ "

ð

; JÞþ"

ð

;

; J;’Þ

:¼

ð

; J; "Þþ"

ð

;

; J;’Þ½22

for a suitable prefixed order p  3; notice that the

nonresonance condition needed to apply averaging

theory is not particularly hard to check sinc e it

involves the unperturbed and completely explicit

Kepler Hamiltonian H

. The idea is now to consider

as a perturbati on of the completely integrable

Hamiltonian

and to apply Kolmogorov’s theo-

rem. Finally, one can check the Kolmogorov’s

nondegenearcy condition, which since

det @

ð

;JÞ

ð

; J

; "Þ¼"

ðdet H

Þdet MþOð"Þ



amounts to check the invertibility of the matrix M.

Such a condition is also checked in Arnol’d (1963b)

with the aid of Le Verrier’s tables and in the

asymptotic regime a

!0.

The spatial three-body problem In order to extend

the previous argument to the spatial case, Arnol’d

suggested connecting the planar and spatial case

through a limiting procedure. Such strategy presents

KAM Theory and Celestial Mechanics 195

analytical problems (the symplectic variables for the

spatial case become singular in the planar limit),

which have not been overcome. However, the

particular structure of the three-body case allows

one to derive a four-degree-of-freedom Hamiltonian,

to which the proof of the planar case can be easily

adapted. The procedure described below is based on

the classical Jacobi’s reduction of the nodes.

First, we inroduc e a convenient set of symplectic

variables. Let, for i = 1, 2, ((L

, G

, 

), (‘

, g

, 

))

denote the Delaunay variables introduced in items

(5) and (6) above: these are the Delaunay variables

associated to the two-body system, Sun–ith planet.

Then, as Poincare´ showed, the variables ((



, 



(



, 



), (

, 

)), where





¼ L





¼ ‘

þ g





ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ðL

 G

cos g





¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ðL

 G

sin g

½23

are symplectic and analytic near circular, non-

coplanar motions; for a detailed discussion of these

and other sets of interesting classical variables, see,

for example, Biasco et al. (2003) and references

therein; the asterisk is introduced to avoid confusion

with a closely related but different set of Poincare´

variables (see below). Let us deno te by

3bp

:¼H

ð0Þ

ð



Þþ"H

ð1Þ

ð



;



;



;



; ;Þ

the Hamiltoni an equation [8] (with n = 2) expressed

in terms of the symplectic variab les

((



, 



), (



, 



), (, )), 



= (



, 



), etc. Recalling

the physical meaning of the Delaunay variables, one

realizes that 

þ 

is the vertical component,

= C  k

, of the total argument C = C

(1)

þ C

(2)

where C

(i)

denotes the angular momentum of the ith

planet with respect to the origin of an inertial

heliocentric frame {k

, k

}. This suggests that the

symplectic variables can be introduced:

ð



;



;



;



; ; Þ¼ð



;



;



;



; ;Þ

with (

, 

):= (

, 

þ 

, 

 

, 

Let



3bp

:¼H

3bp

 

1

denote the Hamiltonian of the spatial three-body

problem in these symplectic variables. Since the

Poisson bracket of 

=

þ 

and H



3bp

vanishes

being an integral for the H

3bp

-flow), the

conjugate angle

is cyclic for H



3bp

, that is,



3bp

¼H



3bp

ð



;



;



;



; 

;

Now (because the total angular momentum C

is preserved), one may restrict attention to the

ten-dimensional invariant (and symplectic) submani-

fold M

ver

defined by fixing the total angular

momentum to be vertical. Such submanifold is

easily described in terms of Delaunay variab les; in

fact, C  k

= 0 = C  k

is equivalent to



 

¼  and G

 

¼ G

 

½24

Thus, M



ver

:= ( M

ver

) is given by



ver

¼ ; 



ð



;



;



;

with



:¼



ð



 H



ð



 H



2



:¼





þ 



Since M



ver

is invariant for the flow 



3bp

(t)   and

 0 for motions starting on



ver

, which implies that (@





3bp



ver

= 0. This

fact allows one to introduce, for fixed values of the

vertical angular momentum 

= c 6¼ 0, the follow-

ing reduced Hamiltonian

red

ð



;



;



;



:¼H



3bp

ð



;



;



;



;



ð



;



;



; cÞ; c;Þ

on the eight-dimensional phase space M

red

:= {



> 0,

 2 T

,(



, 



) 2 B

} endowed with the standard

symplectic form d



^ d



þ d



^ d



being a

ball around the origin in R

). In fact, the (standard)

Hamilton’s equations for H

red

are immediately recog-

nized to be a subsystem of the full (standard)

Hamilton’s equations for H

3bp

when the initial data

are restricted on M



ver

and the constant value of 

chosen to be c. More precisely, if the Hamiltonian flow

of H

red

on M

red

is denoted by 

,then





;



ð



;



;



; cÞ; c;;



¼ 

ðz



Þ;



ðtÞ; c;;

ðtÞ



½25

where we have used the shorthand notations:



= (



,



,



, 



)2M

red

;



(t) =







);

(t) =





3bp

(





(s),c,)ds.Atthispoint,

the scheme used for the planar case may be easily

adapted to the present situation. The nondegeneracy

conditions have been checked in Robutel (1995) where

indications, based on a computer program, have been

given for the validity of the theorem in a wider set of

initial data.

Notice that the dimension of the reduced phase

space of the spatial case is 8, which is also the

dimension of the phase space of the planar case.

196 KAM Theory and Celestial Mechanics