EXPONENTIALLY SMALL ESTIMATES FOR KAM THEOREM
NEAR AN ELLIPTIC EQUILIBRIUM POINT
AMADEU DELSHAMS
Depa amen de Ma ema icaAplicada I, Uni e si a Poli ecnica de Ca alunya
Diagonal 647, E-08028 Ba celona
AND PERE GUTI

ERREZ
Depa amen de Ma ema ica Aplicada II, Uni e si a Poli ecnica de Ca alunya
Pau Ga gal lo 5, E-08028 Ba celona
Abs ac .
We gi e a p ecise s a emen o KAM heo em o a Hamil onian sys em in a
neighb o ho o d o an ellip ic equilib ium p oin . I he equencies o he ellip ic p oin sa is y
a Diophan ine condi ion, wi h exp onen

, and a nondegene acy condi ion is ullled, we
show ha in a neighb o ho o d o adius
he measu e o he complemen o he KAM
o i is exp onen ially small in (1
=
)
1
=
(

+1)
. This esul is ob ained by pu ing he sys em
in Bi kho no mal o m up o an app op ia e o de , and he key p oin elies on gi ing
accu a e es ima es o i s e ms.
1. In o duc ion
We conside a Hamil onian, wi h
n
deg ees o eedom, ha ing he o igin as an ellip ic
equilib ium p oin . In sui able canonical co o dina es, his sys em can b e pu in he o m
H
(
q p
)=
X
s

2
H
s
(
q p
)

(1)
whe e
H
s
is a homogeneous p olynomial o deg ee
s
in (
q p
) o e e y
s

2, and
H
2
(
q p
)=
1
2
n
X
j
=1

j

q
2
j
+
p
2
j

:
(2)
We a e conce ned wi h he exis ence o
n
-dimensional in a ian o i
in a neighbo hood
o he ellip ic p oin . We  s see ha he sys em (1{2) is nea ly-in eg able by pu ing i
in
Bi kho no mal o m
up o an app op ia e deg ee
K

4, p o ided he equency
ec o

=(

1
:::
n
) is non esonan up oo de
K
. A quan i a i e e sion o Bi kho
heo em, implici ly con ained in 4], allows us o ob ain es ima es o he no mal o m.
Like in 9], we conside ac ion{angle a iables in a neighb o ho o d o adius
. Assuming
a sui able nondegene acy condi ion, we apply he known KAM heo em and show ha
mos a jec o ies in a neighb o ho o d o adius
lie in in a ian o i: we ge o he ela i e
measu e o hei complemen an es ima e o he yp e
O

(
K
;
3)
=
2

.
We assume ha

sa ises a Diophan ine condi ion: wi h gi en
>n
;
1and
>
0,
j
k


j

j
k
j

8
k
2
Z
n
n
0
g

(3)
403
whe e wew i e
j
k
j
=
P
n
j
=1
j
k
j
j
.Wesay

o b e
 
-Diophan ine.
Ou main con ibu ion
is o show ha in his case he es ima es o he Bi kho no mal o m allowus ocho ose
he deg ee
K
as a unc ion o
, gi ing ise o an exp onen ially small es ima e o he yp e
exp
(
;

1

1
=
(

+1)
)
o he measu e o he complemen o hein a ian se .We ema k ha he es ima es
gi en in 4] do no allow o ob ain he exp onen 1
=
(

+ 1), bu a wo se one. Ne e heless,
we shall see ha an imp o emen o ha es ima es leads o he announced exp onen .
2. Es ima es o he Bi kho no mal o m
Gi en
K

4, assume ha he equency ec o

is non esonan up oo de
K
:
k


6
=0
o
k
2
Z
n

0
<
j
k
j
K
. The well-known
Bi kho heo em
1, 7] s a es ha , in some
neighb o ho o d o he o igin, he e exis s a canonical ans o ma ion 
(
K
)
, nea o he
iden i y map, such ha
H
(
K
)
=
H


(
K
)
is in Bi kho no mal o m up o deg ee
K
:
H
(
K
)
(
q p
)=


I
+
Z
(
K
)
(
I
)+
R
(
K
)
(
q p
)=
h
(
K
)
(
I
)+
R
(
K
)
(
q p
)

(4)
wi h
Z
(
K
)
(
I
)=
X
4

s

K
s
e en
Z
s
(
I
)

R
(
K
)
(
q p
)=
X
s

K
+1
R
(
K
)
s
(
q p
)

(5)
whe e e e y
Z
s
(
I
) (uniquely de e mined) is a homogeneous p olynomial o deg ee
s=
2in
he ac ion a iables
I
j
=
1
2

q
2
j
+
p
2
j

 j
=1
:::n
and e e y
R
(
K
)
s
(
q p
) is a homogeneous p olynomial o deg ee
s
in (
q p
). Since
h
(
K
)
(
I
)is
in eg able, and in a neighb o ho o d o adius
weha e
R
(
K
)
=
O

K
+1

, i u ns ou
ha
H
(
K
)
is a nea ly-in eg able Hamil onian nea he o igin. Howe e , o apply KAM
heo em o
H
(
K
)
we need quan i a i e es ima es o i s e ms.
As in 4], wein o duce he linea change o complex canonical co o dina es
x
j
=
1
p
2
(
q
j
;
ip
j
)
 y
j
=
;
i
p
2
(
q
j
+
ip
j
)
 j
=1
:::n:
No e ha
q
,
p
a e eal i
y
=
ix
.We dene
j
(
x y
)
j
:= max
j
=1
:::n
q
j
x
j
j
2
+
j
y
j
j
2
.Gi en
>
0,
he eal and complex p olydisks o adius
cen e ed a he o igin will b e deno ed
B
and
b
B
, esp ec i ely.
Fo a gi en homogeneous p olynomial
s
(
x y
)=
X
j
l
+
m
j
=
s
lm
x
l
y
m
,we dene he no m
k
s
k
:=
X
j
l
+
m
j
=
s
j
lm
j
(we use he no a ion
x
l
=
x
l
1
1

x
l
n
n
,
y
m
=
y
m
1
1

y
m
n
n
).
404
P op osi ion 1
Le
H
(
x y
)=
P
s

2
H
s
bea eal Hamil onian wi h
H
2
=


I
, and
assume ha
k
H
s
k
c
s
;
2
d
o
s

3
. Gi en
K

4
, assume ha
j
k


j

K
8
k
2
Z
n

0
<
j
k
j
K
(6)
wi h
0
<
K

1
. Then, he e exis s a eal canonical ans o ma ion

(
K
)
,nea o he
iden i y map, such ha
H
(
K
)
=
H


(
K
)
is in he Bi kho no mal o m
(4{5)
up o
deg ee
K
. Wi h some cons an s
c
1
,
c
2
, one has:
a)
kZ
s
k
c
2
c
s
;
2
1
(
s
;
2)!

3


s
;
1
o
4

s

K
and
s
e en.
b)



R
(
K
)
s




c
2
c
s
;
2
1
(
K
;
3)!(
K
;
2)
s
;
K
+1

3


K
;
1

s
;
K
+1
K
o
s

K
+1
.
c)
The ans o ma ion

(
K
)
is analy ic on
b
B

K

whe e we dene

K
:=

K
c
1
K
:
These es ima es ely on he esul s ob ained in 4], al hough a di ec applica ion would
gi ewo se es ima es, wi h

s
;
3
K
ins ead o

3


s
;
1
in he denomina o s. The imp o e-
men comes om he ac ha , in he cons uc ion o he no mal o m, he only small
di iso s which app ea up o he ob ainmen o
Z
s
co esp ond o he o de s 3
:::s
;
1.
This is c ucial in o de o ge he igh exponen in he es ima es gi en in he las sec ion.
3. Applying KAM heo em
We  s ecall a usual s a emen o KAM heo em. Le us conside a nea ly-in eg able
Hamil onian w i en in ac ion{angle a iables
H
(
 I
)=
h
(
I
)+
(
 I
)

wi h

2
T
n
and
I
2G 
R
n
.To show ha mos o he a jec o ies o
H
lie in
n
-dimensional in a ian o i, one usually imp oses one o he ollowing nondegene acy con-
di ions on he equency map
!
=
h
:
de

@!
@I
(
I
)

6
=0 o de

@!
@I
(
I
)
!
(
I
)
!
(
I
)
>
0
!
6
=0
o e e y
I
2G
.We call hese condi ions
Kolmogo o nondegene acy
and
isoene ge ic
nondegene acy,
esp ec i ely.
We deno e
V

(
G
) a complex neighb o ho o d o adius
a ound
G
.
Theo em 2 (KAM heo em)
Conside he Hamil onian
H
=
h
(
I
)+
(
 I
)
, analy ic
o

2
T
n
and
I
2V

(
G
)
, wi h
o size
"
. Assume ha
!
=
h
is Kolmogo o o
isoene ge ical ly nondegene a e on
G
.Le
>
0
gi en. Fo some cons an s
C
1
,
C
2
,
C
3
,i
"

C
1

2
 

C
2

hen he e exis s
I
T
n
G
l led wi h
n
-dimensional in a ian o i o
H
, sa is ying
mes (
T
n
G
)
nI
]

C
3
(diam
G
)
n
;
1
:
(7)
405
Fo mo e de ailed s a emen s and p o o s, see 8, 9, 2, 3, 5]. Gi en

, i u ns ou ha he
in a ian o i o
H
come om in a ian o i o he unp e u b ed sys em
h
wi h equencies
sa is ying a Diophan ine condi ion o he yp e (3), wi h a xed

. Cho osing


p
"
, he
measu e o he complemen in (7) b ecomes
O
(
p
"
).
Now, ou aim is o apply KAM heo em o he Hamil onian
H
(
K
)
=
h
(
K
)
+
R
(
K
)
in o duced in (4{5). We pu his Hamil onian in ac ion{angle a iables byin o ducing he
known canonical change
q
j
=
q
2
I
j

cos

j
 p
j
=
q
2
I
j

sin

j
 j
=1
:::n:
Howe e , KAM heo em canno b e applied in a di ec way b ecause he change o ac ion{
angle a iables is no analy ic a he hyp e planes
I
j
= 0. Like in 9], his ac o ces
us o emo e a neighbo hood o hese hyp e planes. Thus, o ob ain in a ian o i in he
neighbo hood
B
,we conside o he ac ion a iables he domain
G

:=
(
I
2
R
n
:
I

2

j
I
j
1

2
2
)

wi h
>
0(we use he no a ion
I

a
o mean ha
I
j

a
o
j
=1
:::n
). Wi h a
sui able choice o

, his educ ion o he domain do es no aec essen ially he measu e
es ima es gi en in he nex p op osi ion.
To apply KAM heo em o
H
(
K
)
,we also ha e o equi e ha he equency map
!
(
K
)
=
h
(
K
)
is Kolmogo o o iso ene ge ically nondegene a e. In ac we only assume he
nondegene acy a he o igin i sel , since his suces o ensu e i in a small neighbo hood.
The condi ion we imp ose in ol es he ec o

and he ma ix
A
:=
@
2
Z
4
@I
2
:
Ne e heless, wepoin ou ha highe o de condi ions a e also p ossible.
P op osi ion 3
In he same si ua ion o p oposi ion 1, assume also
de
A
6
=0
o
de

A 

>
0
!
6
=0
:
(8)
Le

K
denedasinpa (c) o p oposi ion 1. Fo some cons an s
c
3
,
c
4
,i
0
<

c
3

K

(9)
hen he e exis s
T
(
K
)
B
l led wi h in a ian o i o
H
(
K
)
, sa is ying
mes
h
B
nT
(
K
)
i

c
4

7

K
!
(
K
;
3)
=
2

mes
B
:
(10)
This esul is ob ained applying KAM heo em on he domain
V

(
G

), wi h
"

K
+1
and


(
K
+1)
=
2
. In ac , his is a mo e elab o a ed e sion o a esul gi en in 9], whe e
a measu e es ima e like (10) is ob ained, also wi h he exp onen (
K
;
3)
=
2.
406
4. The Diophan ine case
Finally,we assume ha he equency ec o

sa ises he Diophan ine condi ion (3) wi h
gi en

and

.We hen ake

K
=

K

in (6) and hence condi ion (9) is ullled i we
cho ose
K

(
=
)
1
=
(

+1)
, leading o an exp onen ially small es ima e.
Theo em 4
Le
H
(
x y
)=
P
s

2
H
s
bea eal Hamil onian wi h
H
2
=


I
, and assume
k
H
s
k
c
s
;
2
d
o
s

3
. Assume ha

is
 
-Diophan ine, wi h
>n
;
1
and
>
0
.
Assume also one o he nondegene acy condi ions
(8)
.Fo some cons an s
c
5
,
c
6
,
c
7
,i
0
<

c
5

hen he e exis s
T
B
l led wi h in a ian o i o
H
, sa is ying
mes 
B
n
T
]

c
6
exp
(
;

c
7


1
=
(

+1)
)

mes
B
:
A ela ed esul has b een announced in 6] whe e, o a xed KAM o us o a nea ly-
in eg able Hamil onian, i is shown ha in a neighb o ho o d o adius
he e exis many
in a ian o i, and he measu e o hei complemen is exp onen ially small in 1
=
.
Acknowledgemen s
This wo k has b een pa ially supp o ed by he EC g an ERBCHRXCT940460. Resea ch
by Amadeu Delshams is also pa ially supp o ed by he spanish g an DGICYT PB94{
0215 and he ca alan g an CIRIT GRQ93{1135, and esea chbyPe e Gu ie ez is also
pa ially supp o ed by he U.P.C. g an PR9409.
Re e ences
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Dynamical sys ems"
.
Am. Ma h. Soc. Col loq. Publ.
9
. Ame ican Ma hema ical
So cie y, New Yo k.
2. H. W. B o e and G. B. Hui ema (1991). A p o o o he iso ene ge ic KAM{ heo em om he `o dina y'
one".
J. Di. Eq.
90
, 52{60 .
3. A. Delshams and P. Gu ie ez (1995). Eec i e s abili y and KAM heo y". Submi ed o
J. Di. Eq.
4. A. Gio gilli, A. Delshams, E. Fon ich, L. Galgani and C. Simo (1989). Eec i e s abili y o a Hamil-
onian sys em nea an ellip ic equilib i um p oin , wi h an applica ion o he es ic ed h ee b o dy
p oblem".
J. Di. Eq.
77
, 167{198 .
5. P.Gu ie ez (1995).
Es abili a e ec i a i o s in a ian s de sis emes hamil onian s quasi-in eg abl es"
.
Do c o al hesis, Uni e si a de Ba celona.
6. A. Mo bidell i and A. Gio gill i (1994). Sup e exp onen i al s abili y o KAM o i". To app ea in
J. S a .
Phys.
7. J. Mose (1968). Lec u es on Hamil onian sys ems".
Memoi s Am. Ma h. Soc.
81
, 1{60 .
8. A. I. Neish ad (1982). Es ima es in he Kolmogo o heo em on conse a ion o condi ional l y p e io dic
mo ions".
J. Appl. Ma h. Mech.
45
, 766{772 .
9. J. Poschel (1982). In eg abili y o Hamil onian sys ems on Can o se s".
Comm. Pu e Appl. Ma h.
35
, 653{696 .