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

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Open Problems

Problem 1 Extend results of Theorems 5 and 6 to

the whole domain of dependence, for small data sets.

The results of Theorems 5 and 6 give a

satisfactory description of gravitational radiation of

general classes of asymptotically flat initial data sets

outside the domain of dependence of a sufficiently

large compact set. It would be desirable to extend

these results to the whole domain of dependence of

initial data sets which satisfy an additional global

smallness assumption similar to that of Theorem 4.

Problem 2 Is strong peeling (and implicitly asymp-

totic simplicity) consistent with physically relevant

data? If not, is weak peeling a good substitute?

Damour and Christodoulou (2000) have given

conclusive evidence that unde r no-incoming-

radiation condition the future null infinity cannot

be smooth. In fact,  = O(r

4

log r)asr !1.

Problem 3 Can one weaken the AF conditions to

include, for example, initial data sets with infinite

ADM angular momentum?

It is reasonable to expect a global stability of

Minkowski result for small initial data sets which

verify, for arbitrarily small ,

g  1 þ 2



 ¼ 0ðr

1c

Þ; k ¼ 0ðr

2c

One expect s in this case that the top null components

 and  decay only like O(r

3

)asr !1along the

null hypersurfaces C(u). It seems that the methods of

Lindblad–Rodnianski can treat this case but can only

give decay estimates for ,  of the form O(r

3þc

Problem 4 Is the Kerr solution in the exterior of

the black hole stable?

The problem remains wide ope n.

See also: Asymptotic Structure and Conformal Infinity;

Classical Groups and Homogeneous Spaces; Critical

Phenomena in Gravitational Collapse; Einstein

Equations: Exact Solutions; Geometric Analysis and

General Relativity; Supergravity.

Further Reading

Bruhat YC (1952) Theoreme d’existence pour certain systemes

d’equations aux derivee partielles nonlineaires. Acta Mathe-

matica 88: 141–225.

Bruhat YCh and Geroch RP Global aspects of the Cauchy

problem in general relativity. Communications in Mathema-

tical Physics 14: 329–335.

Christodoulou D (2000) The Global Initial Value Problem in

General Relativity, Lecture Given at the Ninth Marcel

Grossman Meeting. 2–8 July, Rome.

Christodoulou D and Klainerman S (1990) Asymptotic properties

of linear field equations in Minkowski space. Communications

in Pure and Applied Mathematics XLIII: 137–199.

Christodoulou D and Klainerman S (1993) The global non-linear

stability of the Minkowski space. Princeton Mathematical Series.

Chrus

ciel PT and Delay E (2002) Existence of non-trivial,

vacuum, asymptotically simple spacetimes. Classical and

Quantum Gravity 19: L71–L79.

Corvino J (2000) Scalar curvature deformation and a gluing

construction for the Einstein constraint equations. Commu-

nications in Mathematical Physics 214: 137–189.

Friedrich H (2002) Conformal Einstein evolution. In:

Frauendiener J and Friedrich H (eds.) The Conformal

Structure of Space-Time, Lecture Notes in Physics. Springer.

Hughes T, Kato T, and Marsden J (1976) Well-posed quasilinear

second-order hypebolic systems. Archives for Rational and

Mechanical Analysis 63: 273–294.

Hawking SW and Ellis GFR (1973) The large scale structure of

spacetime. Cambridge Monographs on Mathematical Physics.

Klainerman S (1983) Long-Time Behavior of Solutions to Non-

linear Wave Equations Proc. ICM Warszawa.

Klainerman S (1986) The null condition and global existence to

non-linear wave equations. Lectures in Applied Mathematics

23: 293–326.

Klainerman S and Nicolo` F (1999) On local and global aspects of

the Cauchy problem in general relativity. Classical and

Quantum Gravity 16: R73–R157.

Klainerman S and Nicolo` F (2003a) The evolution problem in

general relativity. Progress in Mathematical Physics, vol. 25.

Boston: Birkhau¨ ser.

Klainerman S and Nicolo` F (2003b) Peeling properties of

asymptotically flat solutions to the Einstein vacuum equations.

Classical and Quantum Gravity.

Lindblad H and Rodnianski I (2003) The weak null condition for

Eisnstein’s equations. Comptes Rendus Hebdomodaires des

Seances de l’Academic des Sciences, Paris 336(11): 901–906.

Lindblad H, and Rodnianski I Global existence for the Einstein

vacuum equations in wave coordinates. Annals of Mathe-

matics (submitted).

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 (1979) Singularities and time asymmetry in general

relativity – an Einstein centenary survey. In: Hawking S and

Israel W (eds.) Cambridge: Cambridge University Press.

Penrose R (1962) Zero rest mass fields including gravitation:

asymptotic behavior. Proceedings of the Royal Society of

London. Series A 270: 159–203.

Wald R (1984) General Relativity. University of Chicago Press.

Stability of Minkowski Space 19

Stability Problems in Celestial Mechanics

A Celletti, Universita

di Roma ‘‘Tor Vergata,’’ Rome,

Italy

Introduction

The long-term stability of planets and satellites might

be desumed by the regular dynamics that we

constantly observe. However, the ultimate fate of

the solar system is an intriguing question, which has

puzzled scientists since antiquity. In the past cen-

turies, the common belief of a regular motion of the

main planets was strengthened by the discovery of a

simple law, due to J D Titius and J E Bode (eight-

eenth century), which provides a recipe to compute

the approximate distances of the planets from the

Sun. Adopting astronomical units as a measure of the

distance, the Titius–Bode law can be stated as

¼ 0:4 þ 0:3  2

AU ½1

where the index n must be selected as provided in

Table 1, which compares the distances computed

according to [1] with the observed values. Titius and

Bode already noticed that it was necessary to skip

one unit in n from Mars to Jupiter; indeed, the

quantity d

= 2.8 AU might correspond to an aver-

age distance of some minor bodies of the asteroid

belt, which had been discovered since the beginning

of the nineteenth century. The studies of the N-body

problem, namely the dynamics of N mutually

attracting bodies (according to Newton’s law),

inspired several mathematical and physical theories:

from the development of perturbation methods to

the discovery of chaotic systems, as attested by the

masterly work of H Poincare´ (1892). In particular,

perturbation theory had relevant applications in

celestial mechanics; for example, it led to the

prediction of the existence of Neptune in the

nineteenth century by J C Adams and U Leverrier

and later to the discovery of Pluto by C Tombaugh,

as a result of unexplained perturbations on Uranus

and Neptune, respectively. Modern advances in

perturbation theories have been provided by the

Kolgomorov–Arnol’d–Moser (KAM) and Nekhor-

oshev theorems, which find broad applications in

celestial mechanics insofar as simple model pro-

blems are concerned.

The stability of the solar syst em can also be

approached through numerical investigations, which

allow one to predict the motion of the celestial

bodies using more realistic models. The results of

the numerical integrations undermine in some cases

the apparent regularity of the solar syst em: in the

following sections, we shall review many examples

of regular and chaotic motions in different con texts

of celestial mechanics, from the N-body problem to

the rotational dynamics.

The Restricted Three-Body Problem

Let P

, ...,P

be N bodies with masses m

, ..., m

which interact through Newton’s law. Let u

(i)

, i = 1, 2, ..., N, denote the position of the bodies

in an inertial reference frame. Normalizing the

gravitational constant to 1, the equations of motion

of the N-body problem have the form

ðiÞ

¼

j¼1; j6¼i

ðu

ðiÞ

 u

ðjÞ

ðiÞ

 u

ðjÞ

; i ¼ 1; ...; N ½2

In the case N = 2, one reduces to the two-body

problem, which can be explicitly solved by means of

Kepler’s laws as follows. Consider, for example, the

Earth–Sun case: for negative values of the energy,

the trajectory of the Earth is an ellipse with one

focus coinciding with the barycenter, which can

practically be identified with the Sun; the Earth–Sun

radius vector describes equal areas in equal times;

the cube of the semimajor ax is is proportional to the

square of the period of revolution.

Consider now an extension to the study of three

bodies such that in the K eplerian approximation P

and P

move around P

and such t hat the

semimajor axi s of P

is greater than that of P

(an

example is obtained identifying P

with the Sun, P

with the Jupiter, and P

with an asteroid of the

main belt). The three-body problem is described by

[2] setting N = 3; a special case is given by the

restricted three-body problem, which describes the

evolution of a ‘‘zero-mass’’ body under the gravita-

tional attraction exerted by an assigned two-body

system. Setting N = 3andm

= 0in[2],the

Table 1 Tititus–Bode law and observed data

Index n

(of [1])

Distance computed

from [1]

Observed

distance (AU)

Mercury 1 0.4 0.39

Venus 0 0.7 0.72

Earth 1 1 1

Mars 2 1.6 1.52

Jupiter 4 5.2 5.2

Saturn 5 10 9.54

Uranus 6 19.6 19.19

20 Stability Problems in Celestial Mechanics

equations governing the restricted three-body pro-

blem are given by

ð1Þ

¼

ðu

ð1Þ

 u

ð2Þ

ð1Þ

 u

ð2Þ

¼

ðu

ð2Þ

 u

ð1Þ

ð2Þ

 u

ð1Þ

ð3Þ

¼

ðu

ð3Þ

 u

ð1Þ

ð3Þ

 u

ð1Þ



ðu

ð3Þ

 u

ð2Þ

ð3Þ

 u

ð2Þ

The first two equations concern the motion of the

primaries P

and P

and they correspond to a

Keplerian two-body problem, whose solution can

be inserted in the equation for u

(3)

, which becomes a

periodically forced second-order equation. The

restricted three-body problem can be conveniently

described in terms of suitable action-angle coordi-

nates, known as Delaunay variables. The present

discussion is rest ricted to the planar case, namely we

assume that the motion of the three bodies takes

place on the same plane. The corresponding Delau-

nay variables, say (L, G, ‘, ) 2 R

 T

, are defined

as follows (Szebehely 1967). Let a and e be,

respectively, the semimajor axis and the eccentricity

of the osculating orbit of P

and let  = 1=m

2=3

; then

Delaunay’s action variables are given by

L ¼ 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

; G ¼ L

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  e

Next, introduce the angle variables: we denote by 

and ’ the longitudes of Jupiter and of the asteroid;

let  be the argument of perihelion, namely the angle

formed by the periapsis direction with a preassigned

reference line, and let u denote the eccentric

anomaly, which can be defined through

tan

’  

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ e

1  e

tan

½3

Let ‘ be the mean anomaly, which is related to the

eccentric anomaly by means of Kepler’s equation

‘ ¼ u  e sin u ½4

Delaunay’s angle variables are repre sented by the

mean anomaly ‘ and by the argument of perihelion

. For completeness, it should be remarked that

the distance r between the minor body P

and the

primary P

is related to the longitude and to the

eccentric anomaly by means of the relations

r ¼

að1  e

1 þ e cosð’  Þ

¼ að1  e cos uÞ½5

In a reference frame centered at one of the

primaries, say P

, let H = H(L, G, ‘, , ) denote

the Hamiltonian function descr ibing the planar

problem; notice that H(L, G, ‘, , ) has two degrees

of freedom and an explicit time dependence through

the longitude  of P

. If the primaries are assumed to

move in circular orbits around thei r common center

of mass, the Hamiltonian function reduces to two

degrees of freedom, where a new variable g is

introduced as the difference between the argument

of perihelion  and the longitude  of the primary.

Normalizing the units of measure so that the

distance between the primaries and the sum of

their masses is unity, the Hamiltonian function H

describing the circular, planar, restricted three-body

problem is given by

HðL; G;‘;gÞ¼

 G þ "FðL; G;‘;gÞ½6

where " = m

. The perturbing function takes the

form

F ¼ r cosðf þ gÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ r

 2r cosðf þ gÞ

where f = ’   represents the true anomaly, namely

the angle formed by the instantaneous orbital radius

with the periapsis line. Notice that the quantities r

and f are functions of the Delaunay variables

through the relations [3]–[5]. As a consequence,

one can expand the perturbing function in the form

(Delaunay 1860)

FðL; G;‘;gÞ¼

j; k0

ð‘; gÞe

where F

are cosine terms with arguments given by

a linear combination of the variables ‘ and g. For

example, the first few terms of the series develop-

ment are given by the following expression:

FðL; G;‘;gÞ¼1 



cos ‘





cosð‘ þ gÞ

e cosð‘ þ 2gÞ





cosð2‘ þ 2gÞ



e cosð3‘ þ 2gÞ



128



cosð3‘ þ3gÞ



cosð4‘ þ4gÞ



128

cosð5‘ þ 5gÞþ ½7

Stability Problems in Celestial Mechanics 21

where the eccentricity is a function of the actions

through e =

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  G

. We remark that the

Hamiltonian [6] is nearly integrable with perturbing

parameter "; indeed, for " = 0 one recovers the two-

body problem describing the interaction between P

and P

, which can be explicitly solved according to

Kepler’s laws.

KAM Stability

Classical perturbation theory , as developed by

Laplace, Lagrange, Delaunay, Poincare´, etc., does

not allow investigation of the stability of the N-body

problem, since the series defining the solution are

generally divergent. In order to justify this state-

ment, let us start by rewriting the unperturbed

Hamiltonian in [6] as

hðL; GÞ¼

 G ½8

so that [6] becomes H(L, G, ‘, g) = h(L, G) þ

"F(L, G, ‘, g). In order to remove the perturbation

to the second order in the perturbing parameter, one

looks for a change of variables (L, G, ‘, g) !

, G

, ‘

, g

) close to the identity, that is,

L ¼ L

þ "

@

@‘

ðL

; G

;‘;gÞ

G ¼ G

þ "

@

ðL

; G

;‘;gÞ

‘

¼ ‘ þ "

@

ðL

; G

;‘;gÞ

¼ g þ "

@

ðL

; G

;‘;gÞ

where (L

, G

, ‘, g) is the generating function of the

transformation. Let

ðL; GÞ¼

 !ðLÞ

In order to perform a first-order perturbation

theory, we look for a generating function

(L

, G

, ‘, g), such that the transformed Hamilto-

nian is integrable up to O("

), namely

þ "

@

@‘

ðL

; G

;‘;gÞ; G

þ "

@

ðL

; G

;‘;gÞ



þ "FL

þ "

@

@‘

ðL

; G

;‘;gÞ; G



þ"

@

ðL

; G

;‘;gÞ;‘;g



¼ h

ðL

; G

Þþ"!ðL

@

@‘

ðL

; G

;‘

; g





@

ðL

; G

;‘

; g

ÞþFðL

; G

;‘

; g



þ Oð"

where h

, G

) is the new unperturbed Hamilto-

nian. If we denote by F

, G

) the average of the

perturbing function over the angle variables, the

new unperturbed Hamiltonian takes the form

ðL

; G

Þ¼hðL

; G

Þþ"F

ðL

; G

Expanding F in Fourier series as F(L, G, ‘, g) =

n, m2Z

(L, G)e

i(n‘þmg)

, the generating function

is given by the following expression:

ðL

; G

;‘;gÞ¼i

n; m2Znf0g

ðL

; G

!ðL

Þn  m

iðn‘þmgÞ

The occurrence of small divisors of the form

!ðLÞn  m

; n; m 2 Z

might prevent the convergence of the series defining

the generating function. In particular, we remark

that zero divisors occur whenever !(L) = m=n. This

situation, which is called an m : n orbit–orbit

resonance, implies that during a given interval of

time the body P

makes m revolutions, whereas P

makes exactly n orbits about P

The control of the occurrence of the small divisors

was obtained through a theorem by A N Kolmogorov,

who made a major breakthrough in the study

of nearly integrable systems. He proved, under

general assumptions, that some regions of the

phase space are almost filled by maximal invar iant

tori. The theorem provides a constructive algorithm

to give estimates on the perturbing parameter,

ensuring the existence of some invariant surfaces.

Kolmogorov’s theorem was later extended by

V I Arnol’d and J Moser, giving rise to the so-called

KAM theory. More precisely, the KAM theorem

can be stated as follows (see, e.g., Arnol’d et al.

(1997)): consider a real-analytic, nearly integrable

Hamiltonian function and fix a rationally indepe n-

dent frequency vector !; if the unperturbe d

Hamiltonian is not degenerate and if the frequency

satisfies a strong nonresonance assumption (called

the diophantine condit ion), for sufficiently small

values of the perturbing parameter, there exists an

invariant torus on which a quasiperiodic motion

with frequency ! takes place. A preliminary

investigation of the stability of the N-body problem

by means of KAM theory (Arnol’d et al. 1997)

leads to the existence of large regions filled by

quasiperiodic motions, provided the masses of the

planets are sufficiently small. Arnol’d’s version of

KAM theorem has been applied by J Laskar and P

Robutel to the spatial three-body planetary problem

(the planetary problem concerns the study of the

22 Stability Problems in Celestial Mechanics

dynamics of two bodies with comparable masses,

moving in the gravitational field of a larger primary)

and the existence of quasiperiodic motions has been

proved for values of the ratio of semimajor axis less

than 0.8 and for inclinations up to 1



Concrete estimates on the strength of the perturba-

tion were given by M He´non: in the context of the

three-body problem, the application of the original

version of Arnol’d’s theorem allows one to prove the

existence of invariant tori for values of the perturbing

parameter (representing the Jupiter–Sun mass ratio)

10

333

while the implementation of Moser’s theo-

rem provides an estimate of 10

50

.Weremarkthatthe

astronomical value of the Jupiter–Sun mass ratio

amounts to 10

3

, showing a relevant discrepancy

between KAM results and physical measurements.

More recently, KAM estimates have been refined and

adapted to the study of significant problems of celestial

mechanics (Celletti and Chierchia 1995). Strong

improvements have been obtained combining accurate

estimates with a computer-assisted implementation,

where the computer is used to perform long computa-

tions concerning the development of the perturbing

series and the check of KAM estimates. The numerical

errors are controlled through the implementation of a

suitable technique, known as interval arithmetic. In

the framework of the planar, circular, restricted three-

body problem, the stability of some asteroids has been

proved by A Celletti and L Chierchia for realistic

values of the perturbing parameter (e.g., for " = 10

3

A suitable approximation of the disturbing function

(namely, a finite truncation of the series development

as in [7]) has been considered. The result relies on an

implementation of a computer-assisted isoenergetic

KAM theorem and on the following remark: in the

four-dimensional phase space, on a fixed energy level

the invariant two-dimensional surfaces separate the

phase space, providing the stability of the actions for

all motions trapped between any two invariant tori.

Since the action variables are related to the semimajor

axis and to the eccentricity of the orbit, one obtains

that the elliptic elements remain close to their initial

values.

A computer-assisted KAM theorem has been

applied by A Giorgilli and U Locatelli to the

planetary (Jupiter–Saturn) problem. Using a suitable

secular approximation, it can be shown that this

model admits two invariant tori, which bound the

orbits corresponding to the initial data of Jupiter

and Saturn.

Nekhoroshev Stability

A diff erent approach in order to study the stability

of nearly inte grable systems is provided by

Nekhoroshev’s theorem (see, e.g., Arnol’d et al.

(1997)), which guarantees, under smallness require-

ments, the stability of the motions on an open set of

initial conditions for exponentially long times.

Consider a Hamiltonian function of the form

Hðy; xÞ¼hðyÞþ"f ðy; xÞ; ðy; xÞ2B  T

½9

where B is an open subset of R

. We assume that h

and f are analyt ic functions and that the integrable

Hamiltonian h satisfies a geometric condition, called

steepness. We remark that functions such as h(L, G)

in [8] satisfy the steepness condition. For sufficiently

small values of ", Nekhoroshev’s theorem states that

any motion (y(t), x(t)) satisfying Hamilton’s equa-

tions associated with [9] is bounded for a finite (but

exponentially long) time, that is,

kyðtÞyð0Þk  y

; for jtjt

ð"

="Þ

where y

, t

, "

, a,andb are suitable positive

constants.

Nekhoroshev’s theorem can be conveniently

applied to the three-body problem, where it provides

a confinement of the action variables, representing

the semimajor axis and the eccentricity of the

osculating orbit. Interesting applications of

Nekhoroshev’s theorem concern the investigation

of the triangular Lagrangian points in the spatial,

restricted three-body problem. (The Lagrangian

points are five equilibrium positions of the planar,

restricted three-body problem in a synodic reference

frame, which rotates with the angular veloc ity of the

primaries. Two of such positions are called trian-

gular, since the configuration of the three bodies is

an equilateral triangle in the orbital plane.) Effective

estimates were developed by A Giorgilli and

C Skokos, showing the existence of a stability

region around the Lagrangian point L

, large

enough to include some known asteroids. In the

same framework, the exponential stability was

proven by G Benettin, F Fasso´, and M Guzzo for

all values of the mass-ratio parameter, except for a

few values of the reduced mass  up to  ’ 0.038.

Numerical Results

The study of the stability of the N-body problem can

be investigated by performing numerical integrations

of the equations of motion. The dynamics of the

outer planets of the solar system (from Jupiter to

Pluto) has been explored by Sussm an and Wisdom

(1992) using a dedicated computer, the Digital

Orrery. The integration of the equations of motion

was performe d over 845 million years; the results

provided evidence of the stability of the major

planets and a chaotic behavior of Pluto. An

Stability Problems in Celestial Mechanics 23

alternative approach, based on an average of the

equations of motion over fast angles, was adopted

by Laskar (1995), where the perturbing function of

the spatial problem was expanded up to the second

order in the masses and up to the fifth powers of the

eccentricity and the inclination. The dynamics of all

planets (excluding Pluto) was investigated by means

of frequency analysis over a time span ranging from

15 Gyr to þ10 Gyr. The numerical integrations

provided evidence of the regularity of the external

planets (from Jupiter to Neptune), a moderate

chaotic behavior of Venus and the Earth, and a

marked chaotic dynamics of Mercury and Mars.

The computations show that the inner solar system

is chaotic, with a Lyapunov time of 5 Myr, thus

preventing any prediction of the evolution over

100 Myr.

The Spin–Orbit Problem

The dynamics of the bodies of the solar system

results from a combination o f a revolutionary

motion around a primary body and a rotation

about an internal axis. A simple mathemati cal

model describing the spin–orbit interaction can

be introduced as follows. Let S be a triaxial

ellipsoidal satellite, which moves about a central

planet P. We denote by T

rev

and T

rot

the periods of

revolution and rotation. A p : q spin–orbit reso-

nance occurs if

rev

rot

; for p; q 2 N; q 6¼ 0

Whenever p = q = 1, the satellite always points the

same face to the host planet. Most of the evolved

satellites or planets are trapped in a 1 : 1 resonance,

with the only exception of Mercury, which is

observed in a nearly 3 : 2 resonance. In order to

introduce a simple mathematical model which

describes the spin–orbit interaction, we assume that:

1. the satellite moves on a Keplerian orbit around the

planet (with semimajor axis a and eccentricity e);

2. the spin axis is perpendicular to the orbit plane;

3. the spin axis coincides with the shortest physical

axis; and

4. dissipative effects as well as perturbations due to

other planets or satellites are neglected.

We denote by A < B < C the principal moments

of inertia of the satellite and by r and f, respectively,

the instantaneous orbital radius and the true

anomaly of the Keplerian orbit. Let x be the angle

between the longest axis of the ellipsoid and a

preassigned reference line. From standard Euler’s

equations for rigid body, the equation of motion in

normalized units (i.e., assuming that the period of

revolution is 2) takes the form

€

x þ

sinð2x  2f Þ¼0 ½10

where " 

(B  A)=C. This equation is integrable

whenever A = B or in the case of zero orbital

eccentricity. Due to the assumption of Keplerian

motion, both r and f are known functions of the

time. Therefore, we can expa nd [10] in Fourier

series as

€

x þ "

m6¼0; m¼1

; e



sinð2x  mtÞ¼0 ½11

where the coefficients W(m=2, e) decay as

W(m=2, e) / e

jm2j

. A further simplification of the

model is obtained as follows. According to (4), we

neglected the dissipative forces and perturbations

due to other bodies. The most important contribu-

tion is due to the nonrigidity of the satellite,

provoking a tidal torque caused by the internal

friction. The size of the dissipative effects is

significantly small compared to the gravitational

terms. Therefore, we decide to retain in [11] only

those terms which are of the same order or larger

than the average effect of the tidal torque. The

following equation results:

€

x þ "

m6¼0; m¼N

; e



sinð2x  mtÞ¼0 ½12

where N

and N

are suitable integers, which depend

on the physical and orbital parameters of the satellite,

while

W(m=2, e) are suitable truncations of the

coefficients W(m=2, e). We remark that eqn [12] can

be derived from Hamilton’s equations associated

with a one-dimensional, time-dependent, nearly

integrable Hamiltonian function with perturbing

parameter " and a trigonometric disturbing function.

Analytical Results

The phase space associated with [12] admits a

Poincare´ map showing a pendulum-like structure:

the periodic orbits are surrounded by librational

curves and the chaotic separatrix divides the libra-

tional regime from the region where rotational

motions can take place. The three-dimensional

phase space is separated by KAM rotational tori

into invariant regions, providing a strong stability

property for all motions confined between any pair

of KAM rotational tori. Let us denote by P(p=q)a

periodic orbit associated with the p : q resonance; in

the context of the model associated with [12], the

24 Stability Problems in Celestial Mechanics

stability of the periodic orbit P(p=q) is obtained by

showing the existence of two invariant tori

T (!

) and T (!

) with !

< p=q <!

. A refined

computer-assisted KAM theore m has been imple-

mented (Celletti 1990) with the aim of proving the

existence of trapping invariant surfaces. Realistic

estimates, in agree ment with the physical values of

the parameters (namely, the equatorial oblateness "

and the eccentricity e), have been obtained in several

examples of spin–orbit commensurabilities, like the

1 : 1 Moon–Earth interaction or the 3 : 2 Mercury–

Sun resonance.

Concerning Nekhoroshev-type estimates, the

classical D’Alembert problem has been studied by

Biasco and Chierchia (2002).Inparticular,an

equatorial ly symmetric oblate planet moving on a

Keplerian orbit around a primary body has been

investigated; the model does not assume any further

constraint on the spin axis. Although the Hamilto-

nian describing this model is properly degenerate, it

is shown that Nekhoroshev-like results apply to the

D’Alembert problem in the proximity of a 1 : 1

resonance.

Numerical Results

The model introduced in [10]–[12] often represents an

unrealistic simplification of the spin–orbit dynamics.

In particular, assumption (1) implies that secular

perturbations of the orbital parameters are neglected,

whereas the hypothesis (2) corresponds to disregarding

the spin–orbit obliquity, namely the angle formed by

the rotational axis with the normal to the orbital

plane. Due to the presence of an equatorial bulge, the

gravitational attraction of the other bodies of the solar

system induces a torque, resulting in a precessional

motion. It is also important to take into account the

changes of the obliquity angle, whose variations

might affect the climatic behavior.

A realistic model for the precession and the

variation of the obliquity has been presented by

Laskar (1995). The numerical simulations and the

frequency- map analysis show tha t the E arth’s

obliquity is actually stable, although a large

chaotic region is found in the interval between



and 90



. Since the present obliquity of the

Earth amounts to 2 3.3



, the Earth is outside the

dangerous region. An interesting simulation was

performed to evaluate the role played by the

Moon. Without the Moon, the extent of the

chaotic region would greatly increase, eventually

preventing the birth of an evoluted life. Among

the inner planets, Mars’ obliquity shows larger

chaotic extent, which drives to variations from



to 60



in a few million years. On the contrary,

the external planets do not show significant

chaotic regions and their obliquities are essen-

tially stabl e.

See also: Averaging Methods; Dynamical Systems in

Mathematical Physics: An Illustration from Water Waves;

Gravitational N-Body Problem (Classical); Hamiltonian

Systems: Stability and Instability Theory; KAM Theory

and Celestial Mechanics; Multiscale Approaches;

Stability Theory and KAM.

Further Reading

Arnol’d VI, Kozlov VV, and Neishtadt AI (1997) Mathematical

Aspects of Classical and Celestial Mechanics. Berlin: Springer.

Biasco L and Chierchia L (2002) Nekhoroshev stability for the

D’Alembert problem of celestial mechanics. Atti Accademia

Nazionale Lincei Classe Scienze Fisiche Matematiche Naturali

Rendiconti Lincei (9) Matematica Applicata 13(2): 85–89.

Celletti A (1990) Analysis of resonances in the spin–orbit problem

in celestial mechanics: the synchronous resonance (Part I).

Journal of Applied Mathematics and Physics (ZAMP) 41:

174–204.

Celletti A and Chierchia L (1995) A Constructive Theory of

Lagrangian Tori and Computer-Assisted Applications.

Dynamics Reported, New Series, vol. 4, pp. 60–129. Berlin:

Springer.

Delaunay C (1860) The´orie du Mouvement de la Lune, Me´moires

de l’Acade´mie des Sciences 1, Tome XXVIII, Paris.

Laskar J (1995) Large scale chaos and marginal stability in the

solar system. In: XIth ICMP Meeting (Paris, July 1994),

pp. 75–120. Cambridge, MA: International Press.

Poincare` H (1892) Les Me´thodes Nouvelles de la Me´chanique

Ce´leste. Paris: Gauthier-Villars.

Sussman GJ and Wisdom J (1992) Chaotic evolution of the solar

system. Science 241: 56–62.

Szebehely V (1967) Theory of Orbits. New York: Academic Press.

Stability Problems in Celestial Mechanics 25

Stability Theory and KAM

G Gentile, Universita

degli Studi ‘‘Roma Tre,’’ Rome,

Italy

Introduction

A Hamiltonian system is a dynamical system whose

equations of motions can be written in terms of a

scalar function, called the Hamiltonian of the system:

if one uses coordinates (p, q) in a domain (phase space)

DR

,whereN is the number of independent

variables one needs to identify a configuration of the

system (degrees of freedom), there is a function H(p, q)

such that

p =  @H=@q and

q = @H=@p.Anintegrable

(Hamiltonian) system is a Hamiltonian system which,

in suitable coordinates (A, a) 2AT

,whereAis an

open subset of R

and T = R=2Z is the standard

torus, can be described by a Hamiltonian H

(A),

that is, depending only on A. The coordinates

(A, a) are called action-angle variables. In such a

case the dynamics is trivial: any initial condition

, a

) evolves in such a way that the action

variables are constants of motion (i.e., A(t) = A

for

all t 2 R), while the angles grow linearly in time as

a(t) = a

þ wt, where w = w(A

)  @

)is

called the rotation (or frequency) vector. An

integrable system can be thought of as a collection

of decoupled (i.e., independent) rotators: the entire

phase space AT

is foliated into invariant tori

and all motions are quasiperiodic. Integrable

systems are stable, in the sense that nearby initial

conditions separate at most linearly in time (in

particular, the actions do not separate at all):

mathematically, this is expressed by the fact that

all the Lyapunov exponents are nonpositive.

An example of an integrable system is any one-

dimensional conservative mechanical system, in any

region of phase space in which motions are

bounded. By increasing the number of degrees of

freedom, exhibiting nontrivial integrable systems

can become a difficult task. The problem of studying

the effects of even small Hamiltonian perturbations

on integrable systems and of understanding if the

latter remain stable, in the aforementioned sense,

was considered by Poincare´ to be the fundamental

problem of dynamics. For a long time, it was

commonly thought that all motions could be

reduced to superpositions of periodic motions,

hence to quasiperiodic motions, but at the end of

nineteenth century it was realized by Boltzmann and

Poincare´ that such a picture was too naive, and that

in reality more complicated motions were possible.

As a consequence of this, it became a widespread

belief that, even when starting from an integrable

system, the introduction of an arbitrarily small

perturbation would break integrability.

This belief was strengthened by the work of

Poincare´ (1898), who showed that the series

describing the solution in a perturbation theory

approach are in general divergent. The source of

divergence in perturbation series is the presence of

small divisors, that is, of denominators of the kind

of w  n, where w is the rotation vector that should

characterize the invariant torus (if existent) and n is

any integer vector. Despite this, however, perturba-

tion series (known as Lindstedt series) continued to

be extensively used by astronomers in problems of

celestial mechanics, such as the study of planetary

motions, for the simple reason that they provided

predictions in good agreement with the observa-

tions. But the feeling that the underlying mathema-

tical tools were unsatisfactory persisted.

In fact, the well-known Fermi–Pasta–Ulam

numerical experiment, in 1955, was originally

conceived in the spirit of confirming that integr-

ability would in general be easily lost. Consider a

chain with N harmonic oscillators, with, say,

periodic boundary conditions, coupled with cubic

and quartic two-body potentials, so that the

Hamiltonian is

Hðp; qÞ¼

i¼1

þ Wðq

iþ1

 q

WðxÞ¼





½1

for ,  real parameters and (p, q) 2 R

 R

. One

can introduce new variables such that the Hamilto-

nian, for  =  = 0, can be written as

ðAÞ¼

i¼1

þ !



¼ w  A ½2

for a suitable rotation vector w = (!

, ..., !

) 2 R

(an explicit computation gives !

= 2 sin(k=N)).

Consider an initial condition in which all the

energy is confined to a few modes, that is, A

6¼ 0at

t = 0 only for a few values of k. For  =  = 0, the

system is integrable, so that A

(t) = 0 for all t 2 R

and for all k such that A

(0) = 0. If the system ceases

to be integrable when the perturbation is switched

on, the energy is likely to start to be shared among

the various modes, and after a long enough time has

26 Stability Theory and KAM

elapsed, an equidistribution of the energy among all

modes (thermalization) might be expected. At least

this behavior was expected by Fermi, Pasta, and

Ulam, but it was not what they found numerically:

on the contrary, all the energy seemed to remain

associated with the modes close to the few initially

excited ones.

At about the same time, Kolmogorov (1954)

published a breakthrough paper going exactly in

the opposite direction: if one perturbs an integrable

system, under some mild conditions on the integr-

able part, most of the tori are preserved, although

slightly deformed. A more precise statement is the

following.

Theorem 1 Let an N-degree-of-freedom Hamilto-

nian system be described by an analytic Hamiltonian

of the form

HðA; aÞ¼H

ðAÞþ"f ð A ; aÞ½3

with " a real parameter (perturbation parameter),

fa2-periodic function of each angle variable

(potential or perturbation), and H

(A) satisfying

the nondegeneracy condition det @

(A) 6¼ 0

(anisochrony condition). If w = w(A)  @

(A) is

fixed to satisfy the Diophantine condition

jw  nj >

jnj



8n 2 Z

n 0 ½4

for some constants C

> 0 and >N 1(here

jnj= j

jþþj

j and  denotes the standard

inner product: w  n = !



þþ!



), then

there is an invariant torus with rotation vector w

for " small enough, say for " smaller than some value

depending on C

and  (and on the function f ).

By saying that there is an invariant torus with

rotation vector w, one means that there is an

invariant surface in phase space on which, in

suitable coordinates, the dynamics is the same as in

the unperturbed case, and the conjugation (i.e., the

change of variables which leads to such coordinates)

is analytic in the angle variables and in the

perturbation parameter. One also says that the

torus of an integrable system (" = 0) is preserved

(or even persists) under a small perturbation.

Note that, a posteriori, this proves convergence of

the perturbation series: however, a direct check of

convergence was performed only recently by

Eliasson (1996). Kolmogorov’s proof was based on

a completely different idea, that is, by performing

iteratively a sequence of canonical transformations

(which are changes of coordinates preserving the

Hamiltonian structure of the equations of motion)

such that at each step the size of the perturbation is

reduced. Of course, on the basis of Poincare´’s result,

this iterative procedure cannot work for all initial

conditions (e.g., when w does not satisfy [4]). The

key point in Kolmogorov’s scheme is to fix the

rotation vector w of the torus one is looking for, in

such a way that the small divisors are controlled

through the Diophantine condition [4] and the

exponentially fast convergence of the algorithm.

New proofs and extensions of Kolmogorov’s

theorem were given later by Arnol’d (1962) and by

Moser (1962); hence, the acronym KAM to denote

such a theorem. Arnol’d gave a more detailed (and

slightly different) proof compared to the original

one by Kolmogorov, and applied the result to the

planar three-body problem, thus showing that

physical applications of the theorem were possible.

Moser, on the other hand, proposed a modified

method using a technique introduced by Nash

(which approximates smooth functions with analy-

tical ones) to deal with the case of systems with

finite smoothness.

For fixed small enough ", the surviving invariant

tori cover a large portion of the phase space, called

the Kolmogorov set; the relative measure of the

region of phase space which is not filled by such tori

tends to zero at least as

ﬃﬃﬃ

for " ! 0. A system

described by a Hamiltonian like [3] is then called a

quasi-integrable Hamiltonian system.

The excluded region of phase space corresponds

to the unperturbed tori which are destroyed by the

perturbation: the rotation vectors of such tori are

close to a resonance, that is, to a value w such that

w  n = 0 for some integer vector n, and these are

exactly the vectors which do not satisfy the

Diophantine condition [4] for any value C

subset of phase space of this kind is called a

resonance region.

At first sight, this would seem to provide an

explanation for the results found by Fermi, Pasta,

and Ulam, but this is not quite the case. First, the

threshold value "

depends on N, and goes to zero

very fast as N !1(in general as N!



for some

>0); however, the results of the numerical

experiments apparently were insensitive to the

number N of oscillators. Second, the KAM theorem

deals with maximal tori, that is, tori characterized

by rotation vectors which have as many components

as the number of degrees of freedom, while the

rotation vectors of the numerical quasiperiodic

solutions seem to involve just a small number of

components.

Finally, as an extra problem, the validity of the

nondegeneracy condition for the unperturbed

Stability Theory and KAM 27

Hamiltonian is violated, because the unperturbed

Hamiltonian is linear in the action variables (one

says that the Hamiltonian is isochronous). Recently,

Rink (2001), by continuing the work by Nishida,

showed that in the Fermi–Pasta–Ulam problem it is

possible to perform a canonical change of coordi-

nates such that in the new variables the Hamiltonian

becomes anisochronous: one uses part of the

perturbation to remove isochrony. But the other

two obstacles remain.

Lower-Dimensional Tori

A natural question is what happens to the invariant

tori corresponding to rotation vectors which are not

rationally independent, that is, vectors satisfying n

resonance conditions, such as w  n

= 0 for n

independent vectors n

, ..., n

, with 1  n  N 2

(the case n = N 1 corresponds to periodic orbits

and is comparatively easy); for instance, one can

take w = (!

, ..., !

,0,..., 0) and, by a suitable

linear change of coordinates, one can always make

the reduction to a case of this kind. In particular,

one can ask if a result analogous to the KAM

theorem holds for these tori. Such a problem for the

model [3] has not been studied very widely in the

literature. What has usually been considered is a

system of n rotators coupled with a system with

s = N n degrees of freedom near an equilibrium

point: then one calls normal coordinates the

coordinates describing the latter, and the role of

the parameter " is played by the size of the normal

coordinates (if their initial conditions are chosen

near the equilibrium point). In the absence of

perturbation (i.e., for " = 0), one has either hyper-

bolic or elliptic or, more generally, mixed tori,

according to the nature of the equilibrium points:

one refers to these tori as lower-dimensional tori, as

they represent n-dimensional invariant surfaces in a

system with N degrees of freedom. Then one can

study the preservation of such tori.

One can prove that, in such a case, at least if

certain generic conditions are satisfied, in suitable

coordinates, n angles rotate with frequencies

, ..., !

, respectively, while the remaining N  n

angles have to be fixed close to some values

corresponding to the extremal points of the function

obtained by averaging the potential over the rotating

angles.

The case of hyperbolic tori is easier, as in the case

of elliptic tori one has to exclude some values of " to

avoid some further resonance conditions between

the rotation vector w and the normal frequencies 

(i.e., the eigenvalues of the linearized system

corresponding to the normal coordinates), known

as the first and second Mel’nikov conditions:

w  n  

jnj



8n 2 Z

n 0; 81  k  s

w  n  

 

jnj



8n 2 Z

n 0

81  k; k

 s

½5

Such conditions appear, with the values of the

normal frequencies slightly modified by terms

depending on ", at each iterative step, and at the

end only for values of " belonging to some Cantor

set one can have elliptic lower-dimensional tori.

The second Mel’nikov conditions are not really

necessary, and in fact they can be relaxed as Bourgain

(1994) has shown; this is an important fact, as it

allows degenerate normal frequencies, which were

forbidden in the previous works by Kuksin (1987),

Eliasson (1988),andPo¨ schel (1989).

Similar results also apply in the case of lower-

dimensional tori for the model [3], which represents

sort of a degenerate situation, as the normal

frequencies vanish for " = 0. Again, one has to use

part of the perturbation to remove the complete

degeneracy of normal frequencies.

Quasiperiodic Solutions in Partial

Differential Equations

For explaining the Fermi–Pasta–Ulam experiment,

one has to deal with systems with arbitrarily many

degrees of freedom. Hence, it is natural to investigate

systems which have ab initio infinitely many

degrees of freedom, such as the nonlinear wave

equation, u

 u

þ V(x)u = ’(u), the nonlinear

Schro¨ dinger equation, iu

 u

þ V(x)u = ’(u), the

nonlinear Korteweg–de Vries equation u

þ u

xxx



u = ’(u), and other systems of nonlinear partial

differential equations (PDEs); the continuum limit of

the Fermi–Pasta–Ulam model gives indeed a non-

linear Korteweg–de Vries equation, as shown by

Zabuski and Kruskal (1965). Here (t, x) 2 R [0, ]

if d is the space dimension, and either periodic

(u(0, t) = u(, t)) or Dirichlet (u(0, t) = u(, t) = 0)

boundary conditions can be considered; ’(u)isa

function analytic in u and starting from orders strictly

higher than one, while V(x) is an analytic function of

x, depending on extra parameters 

, ..., 

. Such a

function is introduced essentially for technical rea-

sons, as we shall see that the eigenvalues 

of the

Sturm–Liouville operator @

þ V(x) must satisfy

some Diophantine conditions. If we set V(x) =  2 R

in the nonlinear wave equation, we obtain the Klein–

Gordon equation, which, in the particular case  = 0,

28 Stability Theory and KAM