Eective Stability for Periodically
Perturbed Hamiltonian Systems
A ngel Jorba,1 and Carles Simo2
Departament de Matematica Aplicada I, ETSEIB
Universitat Politecnica de Catalunya
Diagonal 647, 08028 Barcelona (Spain)
2
Departament de Matematica Aplicada i Analisi
Universitat de Barcelona
Gran Via 585, 08007 Barcelona (Spain)
1
Abstract
In this work we present a method to bound the diusion near an elliptic
equilibrium point of a periodically time-dependent Hamiltonian system. The
method is based on the computation of the normal form (up to a certain degree)
of that Hamiltonian, in order to obtain an adequate number of (approximate)
rst integrals of the motion. Then, bounding the variation of those integrals with
respect to time provides estimates of the diusion of the motion.
The example used to illustrate the method is the Elliptic Spatial Restricted
Three Body Problem, in a neighbourhood of the points L45 . The mass parameter
and the eccentricity are the ones corresponding to the Sun-Jupiter case.
1 Introduction
The study of the nonlinear stability of an elliptic equilibrium point of a Hamiltonian
system is a classical and dicult topic. There are mainly two kind of results concerning
this: results of KAM type (perpetual stability on a Cantor set of initial conditions) and
results of Nekhoroshev type (stability for an exponentially long time span, on an open
set of initial conditions). A survey of both kind of methods can be found in Arnol'd.1
In this work we are going to focus on the results of Nekhoroshev type. Our purpose
will be to bound the diusion of the motion near an elliptic equilibrium point of a
Hamiltonian system. The kind of methods we are going to use is very similar to the
ones used by Giorgilli et al.2 and Simo3 for autonomous Hamiltonians.
radius centered at the origin. Let and T be positive numbers, with < 1.
Denition 1 The origin is ( T )-stable i for all initial condition z0 in B , the corresponding solution is in the ball B during a time span of lenght T .
The values of and T depend on the application we are interested in. For the SunJupiter case, the usual values are = 0:9 (we allow a diusion of 10%) and T of the
order of the age of the Solar system.
In the next sections we are going to see an adapted version of the method used by
Simo,3 in order to bound the diusion (near an elliptic equilibrium point) of a timeperiodic Hamiltonian system. For shortness, the methodology will be directly presented
on a concrete example. The one used is the study of the nonlinear stability of the L5
point of the Elliptic Spatial RTBP (from now on, ESRTBP), taking as eccentricity and
mass parameter the ones corresponding to the Sun-Jupiter case, that is, e = 0:048498458
and = 0:95387536 10;3.
2 Expansion of the Hamiltonian
The rst step of the process is to obtain a power expansion of the Hamiltonian around
the equilibrium point. The way to obtain that expansion depends strongly on the form
of the Hamiltonian we are dealing with. In our example, we have used a recurrence
based on the one of the Legendre polynomials.
Now, let us start from the equations of motion of the ESRTBP. They are (see
Szebehely4 )
x00 ; 2y0 = @!
@x y00 + 2x0 = @!
@y z00 + z0 = @!
@z where
+ + (1 ; ) ! = 1 + ecos f = 21 (x2 + y2 + z2 ) + 1 ;
r
r
2
1
2
and r12 = (x ; )2 + y2 + z2 , r22 = (x ; + 1)2 + y2 + z2 . Moreover, e is the eccentricity
of the two primaries and 0 stands for the derivative with respect to the true anomaly f .
Dening the momenta px = x0 ; y, py = y0 + x, pz = z0 and making the (canonic)
change of variables px ! px ; , py ! py + , pz ! pz , x ! x + , y ! y + and
z ! z, where (
) are the (x y) coordinates of an equilibrium point, the Hamiltonian
of the system is
H = pf + 21 (p2x + p2y + p2z ) + 21 (x2 + y2 + z2 ) + ypx ; xpy ;
; 1 + e1cos f 12 (x2 + y2 + z2 ) + x + y + 1r; + r (1)
PS
PJ
2
2
where rPS
= (x ; xS )2 + (y ; yS )2 + z2 , rPJ
= (x ; xJ )2 + (y ; yJ )2 + z2 , being
(xS yS ) = ( ; ; ), (xJ yJ ) = ( ; 1 ; ; ). The term pf is the momentum
corresponding to f , and has been added to obtain an autonomous Hamiltonian.
S
S
J
S
J
S
>From now on, we are going to focus onpthe L5 case (the same results will hold for
L4 , due to the symmetry): xS = 1=2, yS = 3=2.
Our purpose now is to expand this Hamiltonian in a power series of x, px, y, py , z
and pz :
X
H (x px y py z pz f pf ) = pf + Hj (x px y py z pz f )
j 2
where Hj is an homogeneouos polynomial of degree j , whose coecients are Fourier
series with respect to f .
To reduce the global computational eort, we will perform the change of variables
that puts H2 in normal form, at the same time that the Hamiltonian is expanded. For
this reason, the rst point we are going to deal with is the normal form of H2.
2.1 Normal Form of H2
This corresponds to a linear dierential equation with periodic coecients, that can be
reduced to constant coecients by means of the classical Floquet Theorem. We have
obtained the change of variables numerically, and we have performed a Fourier analysis
in order to obtain an (approximate) analytical expression for the change. At the same
time, we want that H2 be of the form:
H2 = !21 (x2 + p2x) + !22 (y2 + p2y ) + !23 (z2 + p2z )
in order to simplify the next steps. This is done by a linear change of variables, that
can be composed with the Floquet one to obtain a single change making both things.
Finally, in order to obtain an adequate expression for the calculus of the normal form
it is convenient to complexify the expansion. The reason is that, in suitable complex
coordinates, H2 takes the form
H2 =
3
X
j =1
p
;1!j qj pj that will be very useful (it will save time, memory and the process will not be ill
conditioned) during the computations. For that purpose we perform the change of
variables
p
p
q
1 + ;1p1
p
x=
px = ;1pq1 + p1 (2)
2
2
and similar expressions for q2, p2, q3 and p3 . The composition of this change with the
previous ones provides the nal change we were looking for.
2.2 Expansion
;1 (r;1 is obtained
In order to expand (1), we are going to focus on the expansion of rPS
PJ
with the same procedure). It is known that
1
X
1
1 =p
=
nP (cos )
rPS
1 ; 2 cos + 2 n=0 n
where is the angle between (xS yS 0) and (x y z), 2 = x2 + y2 + z2 and Pn is the
Legendre polynomial of degree n. Let us dene An as n Pn(cos ) (note that An is
n
n (x2 + y2 + z2 )A An+1 = 2nn++11 (xxS + yyS )An ; n +
n;1
1
(3)
being A0 = 1 and A1 = xxS + yyS . We have used this recurrence to obtain the expansion
of the Hamiltonian, using for x and y the expressions provided by the nal \Floquet"
change. Finally, the remaining terms of (1) are computed directly.
3 Normal form
In what follows, k will be a multiindex, splitted as k = (k1 k2), where k1 and k2
correspond to positions and momenta respectively.
The computation of the normal form is based on the following proposition:
Proposition 1 Let us consider the Hamiltonian
H = pf + H2 (q p) +
P
rX
;1
i=3
Hi(q p) + Hr (q p f ) + Hr+1(q p f ) + P
k k1 k2
where r > 2, H2(q p) = !iqi pi (!i 2 C ), H
(
q
p
)
=
i
j
k
j
=i hi q p , Hr (q p f ) =
P
k
k1 k2 and hk (f ) = P hk exp (jf p;1). Let us dene Gr = Gr (q p f ) =
j
k
j
=r hr (f )q p
j rj
r
P
k (f )q k1 pk2 as follows:
g
jkj=r r
1. if k1 6= k2 :
grk (f ) =
2. if k1 = k2 :
k
X
p
h
ck ; hkr0
rj
p
+
exp
(
jf
;1):
< ! k2 ; k1 > j6=0 j ;1; < ! k2 ; k1 >
grk (f ) =
phrj exp (jf p;1):
j 6=0 j ;1
X
k
Then, the new Hamiltonian H 0 obtained from H by means of the change of variables
given by the generating function Gr ,
H 0 = H + fH Gr g + 2!1 ffH Gr g Gr g + satises that
H 0 = pf + H2 (q p) +
P
rX
;1
i=3
where Hr0 (q p) = (h0 )kr qk1 pk2 and
Hi(q p) + Hr0 (q p) + Hr0 +1(q p f ) + (
1
k2
= chkk ifif kk1 6=
= k2
r0
Remark: The value ck that appears in the function G can be selected according to
several criteria. In our case, as we want to obtain an integrable normal form, we have
chosen that value equal to 0. It is also possible to chose ck = hkr0 for the \quasiresonant"
values of k, to alleviate the eect of the small divisors, but what we obtain in this case
is a (in general non-integrable) seminormal form.
(h0 )k
r
In the elliptical problem, we have a (exact) resonance between the frequency of the true
anomaly and the frequency of the vertical mode of the RTBP. Due to that resonance,
itpis not possible to obtain an autonomous normal form, because some of the divisors
j ;1; < ! k2 ; k1 > appearing in the generating function are zero. In that case, we
k equal to zero.
select the corresponding grj
3.2 Realication
The last step of the procedure is to return to real coordinates and to introduce action
variables, in order to study the dynamics of the nal Hamiltonian. The rst part is
done by using the inverse of the map given in (2). We denote again by
x, px, etc. the
p
real coordinates. The second one by using the Poincare change x = 2I1 cos '1 , px =
;p2I1 sin '1, and similar for the other variables. In this way, the global transformation
preserves the real character of variables and Hamiltonian.
With these changes, we obtain a nal Hamiltonian of the form:
H = N + R(m+1) + If where m is the degree of the normal form and
N = N (I1 I2 I3 '3 ; f )
that depends on '3 ; f because of the 1 : 1 resonance between the true anomaly f and
the angle '3 (corresponding to the action I3) of the vertical mode of the RTBP. Here
If is the actual momentum conjugated to f .
All the previous computations have been eectively carried out up to order 20. The
Fourier coecients which appear in all the process with amplitude less than 10;15 have
been dropped.
3.3 An approximation to the dynamics
To study the ow of this Hamiltonian, let us perform the following change of variables
If = J0 ; J3 I1 = J1 I2 = J2
I3 = J3
f = 0 '1 = 1 '2 = 2 '3 = 3 + 0 to obtain
where
H = N + R(m+1) + J0
N = N (J1 J2 J3 3) = !1J1 + !2J2 + O2(J1 J2 J3)
is an integrable Hamiltonian and R(m+1) = Pj>m Rj . More concretely, up to second
order in the Jj variables, N is given by
N(2) = !1J1 + !2J2 + 11 J12 + 12 J1J2 + 22J22 +
+J3 aJ1 + dJ2 + (bJ1 + eJ2 ) cos 23 + (cJ1 + fJ2) sin 23] + gJ32
where the numerical values obtained for the constants in the current example are given
in Table 1.
The dynamics described by this Hamiltonian is very simple: as J1 and J2 are rst
integrals, we can take J1 = C1, J2 = C2 (C12 are constants) to obtain an one-degree
!1
11
22
b
d
f
= ;.8080513430831042E;01 !2 = .9967588604945699E+00
= .5842667892951852E+00 12 = ;.1551131477053751E+00
= .5641749211325000E;02 a = .5439031817085827E;01
= .8735695861611382E;04
c = .2162237916664492E;03
= .5793623535537272E;02
e = ;.5265749230961082E;05
= ;.7753646646134309E;05 g = ;.1787604308798145E;03
of freedom Hamiltonian. In our case this is, essentially, a pendulum depending on the
parameters J1 and J2. Using N(2) as an approximation, the xed points are located at
q
q
q
cos 23 = n= n2 + p2 sin 23 = p= n2 + p2 J3 = (m n2 + p2)=(;2g)
where m = aJ1 + dJ2, n = bJ1 + eJ2 and p = cJ1 + fJ2. The points with the + sign
are hyperbolic and the ones with the ; sign are elliptic. If J1 = J2 = 0 the angle
3 is moving quite slowly, with limit frequency equal to zero when J3 goes to zero.
When J1 and/or J2 move away from zero a bifurcation occurs. This is related to the
1 to 1 resonance between the vertical mode and the frequency of the elliptic motion.
In particular the existence of normally hyperbolic tori of dimensions 2 and 3 follows.
However, this will not aect the existence of eective stability, as we shall show.
4 Bounding the diusion
To bound the diusion speed it is enough to bound J10 , J20 and N 0 , because N is also
an approximate rst integral of the motion. To do it, we recall that
Jj0 = fJj H g = fJj R(m+1) g j = 1 2 N 0 = fN R(m+1) g
(4)
and this can be bounded easily if we have bounds for kRm+1 k and for the Poisson
bracket.
4.1 Norms
The norm used to produce the bounds mentioned above is
X
X X k
kUr k = kukr (f )k =
jurj j
jkj=r
jkj=r j
where U is an homogeneous polynomial of degree r, ukr are its coecients, and ukrj are
the corresponding Fourier coecients.
This norm has two interesting properties:
1. Let us denote by kUr kB the sup norm of the homogeneous polynomial Ur of
degree r over the ball centered in the origin and with radius , and for all (real)
time f . Then
kUr kB kUr kr :
2. Let Ur and Vs be homogeneous polynomials of degree r and s. Then
kfUr Vsgk rskUr kkVsk:
j B
B
obtained using dierent norms (see Simo3).
4.2 Final results
>From (4), and using the norms mentioned above, it is not dicult to obtain
X
jJj0 j lkRl kl j = 1 2
l>m
where the values kRl k can be estimated of the following form: >From the recurrence (3),
we obtain bounds for the norms of the homogeneous polynomials Hr that appear in the
Hamiltonian expansion. Then, we bound the norms of the homogeneous components
of the successive Hamiltonians obtained (using G3 to G20 ). In particular we get kRl k.
We have used the recurrent relations given by the Lie transforms to obtain the bounds
up to order 800. Note that, with this, we have bounded the diusion with respect to
the planar variables.
Now we would like to bound the diusion in the vertical direction. To do that,
instead of bounding jN 0j, let us dene rst Nf as N minus the part of N depending only
on J1 , J2 (obviously, Nf is also an approximate rst integral).
Now, we bound Nf0 (as it was done with J10 2 by means of (4)). From this bound we
need to derive a bound for the diusion in the vertical direction. At this point we want
to remark a diculty in this approach: J3 is not an approximate rst integral but a
fast variable of the system. The J3 variable can oscillate around the elliptic equilibrium
point mentioned in Section 3.3.
We have used the following approach: to ensure that if we start at a ball of radius
then we end, at most, at the boundary of a ball of radius we should have
min
jNf(zf ) ; Nf(zi)j T jNf0 jB :
(z z )2@B @B
i f
We have used = 0:5 (see later) and T the approximate age of the Solar system
(T = 4:5 109 years 2:4 109 adimensional units). The maximum value of satisfying the previous condition is then obtained.
Notice the following fact about this method: even if the diusion for Nf were exactly
0, we would have \diusion" in the vertical direction, due to the (possible) libration of
J3. Obviously, this diusion is not a real one. Otherwise, this \false" diusion is a fast
phenomenon that can be observed, and it has a physical meaning.
To have an idea of the amount of this diusion, we can compute from the normal
form a bound A of the maximum size of the above-mentioned libration motion. This size
depends on the value J3 for which we have the elliptic equilibrium point (see Section
3.3). It is not dicult to obtain that A=J3 < 0:27 (this follows, essentially, from a
careful analysis of N(2) ). This is the reason why we have taken a small value of (0.5
instead of something close to 1).
Finally, using = 0:5 and T = 2:4 109, as stated before, the radius of the ball of
( T )-stability is found to be = 0:571 10;3.
Now we need to send that ball back to the original coordinates. We are going to
split this process in two steps: the normal form change (the nonlinear part of the change
obtained in Section 3) and the Floquet change.
The deformation produced by the normal form change can be bounded from the
norm of the coecients of this change of variables. This shows that this ball could be
reduced in a factor of 5=6, that is, to the value b = 0:476 10;3.
on the true anomaly f in a periodic way. To obtain a region of eective stability
independent of f we have intersected (for all f ) all these regions. This produces a
domain with a \banana" shape, having a minimum diameter of 0:11b and a maximum
one of 9:8b. This implies that the largest ball contained in that domain and centered
at the origin (L45 ) has a radius of 0:52 10;4 adimensional units. On the other
hand, in some \good" direction, the distance of stability is multiplied by a factor of
9:8=0:11 89.
4.3 Remarks
Numerical simulations of the motion close to L45 in the same problem show that there
exist points close to L45 (but outside the region mentioned before) that go away in a
very short time (see Gomez et al.5# Simo6). This is due to the eect of the 1:1 resonance
between the true anomaly and the vertical mode of the RTBP (this phenomenon is now
being studied (see Jorba and Simo7)). This eect is more evident when the value of is bigger (i.e. Earth-Moon system), because the (angle of) splitting between the stable
and unstable manifolds of the hyperbolic tori mentioned above grows very fast when increases.
There are also other regions of stability that are even larger, allowing for large
values for the inclinations (see Gomez et al.5). These regions are the ones containing
the Trojan asteroids.
Acknowledgements. The authors have been partially supported by the CICYT grant
ESP91-0403.
References
1. V.I. Arnol'd: \Dynamical Systems III", Springer-Verlag (1988).
2. A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simo: Eective stability
for a Hamiltonian system near an elliptic equilibrium point, with an application to
the Restricted Three Body Problem, Journal of Dierential Equations 77:1 (1989).
3. C. Simo: Estabilitat de sistemes Hamiltonians, Mem. de la Real Acad. de Cienc. i
Art. de Barcelona, Vol. XLVIII, no. 7 (1989).
4. V. Szebehely: \Theory of Orbits", Academic Press (1967).
5. G. Gomez, A. Jorba, J. Masdemont and C. Simo: Study of Poincare maps for Orbits near Lagrangian Points, ESOC contract 9711/91/D/IM(SC), Second Progress
Report (1992).
6. C. Simo: Stability regions for the elliptic RTBP near the triangular points (in
progress).
7. A. Jorba and C. Simo: Hyperbolic tori close to the equilateral points of the elliptic
RTBP (in progress).