SYMMETRY AND RESONANCE
IN
HAMILTONIAN SYSTEMS
J. M. Tuwankotta
F. Verhulst
Mathematisch Instituut
PO Box 80.010, 3508 TA Utrecht,
The Netherlands
Novemb er 23, 1999
Abstract
In this paper we study resonances in two degrees of freedom, autonomous, hamil-
tonian systems. Due to the presence of a symmetry condition on one of the degrees of
freedom, we show that some of the resonances vanish as lower order resonances. Af-
ter determining the size of the resonance domain, weinvestigate this order change of
resonance in a rather general p otential problem with discrete symmetry and consider
as an example the Henon-Heiles family of hamiltonians. We also study a classical
example of a mechanical system with symmetry, the elastic p endulum, which leads
to a natural hierarchy of resonances with the 4 : 1-resonance as the most prominent
after the 2 : 1-resonance and which explains why the 3 : 1-resonance is neglected.
Keywords.
Hamiltonian mechanics, higher-order resonance, normal forms, symmetry,
elastic p endulum.
AMS clasication.
34E05, 70H33, 70K30
1 Intro duction
Symmetries play an essential part in studying the theory and applications of dynamical
systems. For a general dynamical systems reference see [3], for symmetry in the context
of hamiltonian systems see [4] and also [1], or [12]. In the older literature, attention
was usually payed to the relation between symmetry and the existence of rst integrals
On leave from Jurusan Matematika, FMIPA,Institut Teknologi Bandung, Ganesha no. 10, Bandung,
Indonesia
1
but recently the relation b etween symmetry and resonance, in particular its inuence on
normal forms has been explored, see [13] for references.
In our analysis we shall be using a small parameter
"
whichisintroduced by re-scaling
the variables. The implication is that, as
"
is small we analyze the dynamics of the
hamiltonian ow in the neighborho od of equilibrium corresponding with the origin of phase-
space. Note that
"
2
is a measure for the energy with resp ect to equilibrium. Putting
"
=0,
the equations of motion reduce to linear decoupled oscillators.
An imp ortant tool to analyze hamiltonian systems are normal forms. Based on the
lowest degree of the expanded hamiltonian function where resonant terms are found, one
can classify the resonances into three classes, i.e. rst order, second order and higher order
resonance. First order and second order resonances of two degrees of freedom have b een
considered intensively in the literature while higher order resonance has been considered in
[6] and [7]. One can also classify resonance in the sense of energy interchange b etween the
degrees of freedom. Terms like strong (or genuine) resonance and weak resonance are used
to express the order of energy interchange on a certain time-scale whichischaracteristic
for the dynamics of the system.
In this paper we are fo cusing on resonance in the presence of symmetry, see [13] for an
introduction. In section 2 we will indicate how symmetry assumptions aect resonance and
the normal forms. We use Birkho-Gustavson normalization which is equivalent with aver-
aging techniques. In section 3 we give a sharp estimate of the size of the resonance domain
at higher order resonance. Section 4 focuses on a special resonance, the 1 : 2-resonance for
symmetric potential problems; we discuss an example from an imp ortant family of poten-
tial problems. Section 5 discusses one of the classical mechanical examples with symmetry,
the elastic pendulum. In this problem, we show that the symmetry assumption pro duces
a new hierarchy of resonances.
2 Higher order resonance triggered by symmetry
Consider the two degrees of freedom hamiltonian
H
(
p
1
;q
1
;p
2
;q
2
)=
1
2
!
1
,
p
2
1
+
q
2
1
+
1
2
!
2
,
p
2
2
+
q
2
2
+
H
3
+
H
4
+
:
(1)
with
H
k
,
k
3 a homogeneous p olynomial of degree
k
. Using action-angle transformation
and Birkho-Gustavson normalization, we can transform the hamiltonian into normal form
while preserving the hamiltonian structure. A hamiltonian
H
is said to b e in Birkho
normal form of degree 2
k
if it can be written as
H
=
!
1
1
+
!
2
2
+
P
2
(
1
;
2
)+
P
3
(
1
;
2
)+
+
P
k
(
1
;
2
)
;
where
P
i
(
1
;
2
) is homogeneous polynomial of degree
i
in
j
=
1
2
(
p
j
2
+
q
j
2
)
;j
=1
;
2. The
variables
1
;
2
are called actions; note that if Birkho normalization is p ossible, the angles
have b een eliminated. If a hamiltonian can be transformed into Birkho normal form, the
2
dynamics is fairly regular. The system is integrable with integral manifolds which are tori
described by taking
1
;
2
constant. The ow on the tori is quasi-p eriodic.
In normalizing, it is convenientif we transform to complex co ordinates by
x
j
=
p
j
+
iq
j
y
j
=
p
j
,
iq
j
;j
=1
;
2
;
with corresponding hamiltonian
~
H
=2
iH
.
The idea of Birkho-Gustavson normalization is to transform
H
(wehave dropped the
tilde) so that the transformed hamiltonian becomes
H
(
x
1
;y
1
;x
2
;y
2
)=
B
k
(
1
;
2
)+
R
(
x
1
;y
1
;x
2
;y
2
) (2)
where
B
k
is in Birkho normal form with
k
as high as possible.
R
is a p olynomial which
has degree of either 2
k
or 2
k
+ 1 in (
x
1
;y
1
;x
2
;y
2
); it contains terms which can not be
expressed in
1
;
2
alone. The terms
R
are also known as resonantinteraction terms and
H
in this form is called the Birkho-Gustavson normal form. In this paper we will refer
to the terms in
R
as resonant terms. For normalization one can use a generating function
or suitable averaging techniques. See for example [1] appendix 8 or [12] chapter 11.
The presence of resonant terms of the lowest degree in the hamiltonian determines
until what order the normalization should be carried out. For example, consider a general
hamiltonian (1) and assume there is a pair of natural numbers (
m; n
) such that
m
n
=
!
1
!
2
where
m
and
n
are relatively prime. The resonant terms of the lowest degree are generally
found in
H
m
+
n
;
!
1
:
!
2
is said to be a lower order resonance if the corresponding resonant
terms of the lowest degree are found in
H
k
with
k<
5.
It turns out that some of the lower order resonances are eliminated by symmetry in
which case
m
and
n
need not b e relative prime. We write the normal form (2) as,
H
(
x
1
;y
1
;x
2
;y
2
)=
B
k
(
1
;
2
)+2
i
(
Dx
1
n
y
2
m
+
Dy
1
n
x
2
m
)+
:
(3)
where
j
=
1
2
x
j
y
j
;j
=1
;
2. In table 1 we present a list of lower order resonances and its
corresponding resonant terms of the lowest degree. The second column shows resonant
terms in a general hamiltonian system while the third column is for a hamiltonian system
with symmetry in the second degree of freedom, i.e.
H
(
p
1
;q
1
;
,
p
2
;
,
q
2
)=
H
(
p
1
;q
1
;p
2
;q
2
).
Except for the 1 : 1 and 2 : 1 -resonances, the other resonances are aected by the
symmetry assumption. For example, the 1 : 2-resonance in the general hamiltonian has
resonant terms of the form
x
2
1
y
2
or
x
2
y
2
1
. These terms vanish because of the symmetry
assumption. Thus, instead of these terms which arise from
H
3
, the resonant terms in the
normal form derive from
H
6
in the form of
x
4
1
y
2
2
or
x
2
2
y
4
1
.Weintro duce the small parameter
"
by rescaling
x
i
=
"
x
i
;y
i
=
"
y
i
and divide the hamiltonian by
"
2
. The normal form of the
hamiltonian in the 1 : 2-resonance with discrete symmetry in the second degree of freedom
is
H
(
x
1
;y
1
;x
2
;y
2
)=
i
[
1
+2
2
]+
"
2
A
1
2
1
+
A
2
1
2
+
A
3
2
2
+
"
4
B
1
3
1
+
B
2
2
1
2
+
B
3
1
2
2
+
B
4
3
2
+
"
4
[
Dx
4
1
y
2
2
+
Dy
4
1
x
2
2
]
+
:
(4)
3
Resonant term
!
1
:
!
2
General Hamiltonian Symmetric in
x
2
;y
2
1:2
x
1
2
y
2
; x
2
y
1
2
x
1
4
y
2
2
; x
2
2
y
1
4
2:1
x
2
2
y
1
; x
1
y
2
2
x
2
2
y
1
; x
1
y
2
2
1:3
x
1
3
y
2
; x
2
y
1
3
x
1
6
y
2
2
; x
2
2
y
1
6
3:1
x
1
y
2
3
; x
2
3
y
1
x
1
2
y
2
6
; x
2
6
y
1
2
1:1
x
1
2
y
2
2
; x
2
2
y
1
2
x
1
2
y
2
2
; x
2
2
y
1
2
x
1
2
y
1
y
2
; x
1
x
2
y
2
2
x
1
y
1
2
x
2
; y
1
x
2
2
y
2
Table 1: The table presents lower order resonant terms which cannot b e removed by
Birkho normalization. The second column shows resonant terms in the general case
while in the third column wehave added the symmetry condition
H
(
x
1
;y
1
;
,
x
2
;
,
y
2
)=
H
(
x
1
;y
1
;x
2
;y
2
).
The constants
D
and
D
are complex conjugate.
It is also clear that symmetry in the second degree of freedom does not aect the 2 : 1-
resonance. If we assume the symmetry is in the rst degree of freedom, then this resonance
will b e aected while the 1 : 2-resonance will not. On the other hand, both the 3 : 1- and
1 : 3-resonances are eliminated as a lower order resonance by the symmetry assumption, no
matter on which degree of freedom the symmetric condition is assumed. As in mechanics
one often has symmetries, this may also explain why these resonances received not much
attention in the literature. This is demonstrated clearly for the elastic p endulum in section
5. For the 1 : 1-resonance, symmetry conditions on any degree of freedom (or even in both)
do not push it into higher order resonance.
3 The Resonance Domain
Consider the normal form of a hamiltonian at higher order resonance
H
=
!
1
1
+
!
2
2
+
"
2
P
2
(
1
;
2
)+
+
"
m
+
n
,
2
(
1
n
2
m
)
1
2
cos(
)
;
(5)
where
=
n'
1
,
m'
2
+
,(
1
;'
1
) and (
2
;'
2
) are the action-angle variables of the rst
and the second degree of freedom, respectively,
m
and
n
are natural numbers satisfying
m
+
n
5 and
m
n
=
!
1
!
2
; and
2
[0
;
2
). Note that
P
k
is a homogeneous p olynomial of degree
k
and it corresponds to the
H
2
k
term in the hamiltonian (1). Independentintegrals of the
system are
I
1
(
1
;
2
)
!
1
1
+
!
2
2
=
E
;
and
I
2
(
1
;
2
;'
1
;'
2
;"
)
(
H
,I
1
(
1
;
2
))
="
2
=
C
.
In a seminal paper [6], Sanders describ ed the ow on the energy manifold as follows. In
the case of higher order resonance interesting dynamics takes place in the resonance domain
whichisimb edded in the energy manifold. The resonance domain contains a stable and
an unstable perio dic solution; the domain is foliated into tori on which the interaction
4
between the two degrees of freedom takes place. In [6] the time-scale of the interaction is
"
,
(
m
+
n
)
=
2
and the size
d
"
of the resonance domain is estimated to b e
O
(
"
(
m
+
n
,
4)
=
6
).
The estimate of
d
"
given in [6] is an upp er limit, it dep ends on the approximation
technique used there. Van den Bro ek [9](pp. 65-67) gavenumerical evidence that the
size of the resonance domain is actually smaller. In this section we shall present a sharp
estimate of the size
d
"
whichwe derive from a Poincare section of the ow.
The derivation runs as follows. First eliminate one of the actions, say
1
using
I
1
. Then
wechoose the section by setting
'
1
=0. Thus wehave a section in the second degree of
freedom direction which is transversal to the ow of the system. For simplicity,we put
= 0 (otherwise we can rotate the coordinate with resp ect to the origin). The second
integral
I
2
then becomes
I
2
(
2
;'
2
;"
)=
I
2
((
E
,
!
2
2
)
=!
1
;
2
;
0
;'
2
;"
). Write
P
(
q ; p; "
)=
I
2
p
2
+
q
2
2
;
arccos
q
p
p
2
+
q
2
!
;"
!
:
(6)
Fixing a value for
E
,we draw the contour plot of (6) which gives us the Poincare map.
The contour plot of
P
in the
q
-
p
plane (for a xed
"
) mainly consists of circles sur-
rounding the origin. This is due to the fact that in the equations of motion, the equation
for the actions are of order
"
m
+
n
,
2
and the equation for
of order
"
2
. This implies that
for most of the initial conditions, the actions are constant up to order
"
m
+
n
,
2
and only the
angles are varying. This condition fails to hold in a region where the righthand side of the
equation for
is zero or becomes small. Up to order
"
2
, the location of this region can be
found by solving
n
@P
2
@
1
,
m
@P
2
@
2
=0
:
Note that in phase space, the equation ab ove denes the so-called resonance manifold. On
this resonance manifold there exist at least 2 short periodic solutions of the system (more
if
m
and
n
are not relatively prime).
In the contour plot, these short perio dic orbits app ear as 2
m
xed points (excluding
the origin) which are saddles and centers corresp onding to the unstable and stable p eriodic
orbit. Eachtwo neighbouring saddles are connected by a heteroclinic cycle. Inside each
domain bounded by these heteroclinic cycles, also known as the resonance domain, there
is a center point. For example, see gure 3 in section 5. We approximate the size of this
domain by calculating the distance between the twointersection points of the heteroclinic
cycle and a straight line
p
=
q
for a
such that a center point is on the line.
Suppose we found one of the saddles (
q
s
;p
s
) and one of the centers (
q
c
;p
c
). We calculate
C
s
"
=
P
(
q
s
;p
s
;"
) and
C
c
"
=
P
(
q
c
;p
c
;"
). Note that since the integral
I
2
depends only on
the actions up to order
"
m
+
n
,
4
wehave
C
s
"
,
C
c
"
=
O
(
"
m
+
n
,
4
). The heteroclinic cycles are
given by the equation
P
(
q ; p; "
)=
C
s
"
and the intersection with the line
p
=
q
is given by
solving
P
(
q; q; "
)=
C
s
"
.Write
q
=
q
c
+
"
;
2
IR
.Wewant to determine
which leads
us to the size of the domain.
Note that
P
(
q; q; "
)=
P
4
(
q; q
)+
"
2
P
6
(
q; q
)+
+
"
m
+
n
,
4
R
(
q; q; "
)
;
where
P
k
is a
non-homogenous (in general) p olynomial of degree
k
and
R
is determined by the resonant
5
