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, in order to determine the other function. 2.2
Shapere-Wilczek's Connection and Curvature Forms
Appealing to the Stokes paradox, a form of which states that the only solution to the Stokes equation corresponding to a rigid translation of a cylinder is a rigid translation of the fluid as a whole, Shapere and Wilczek asserted that the rotation and translation of the organism associated with a vector field V(a) on the boundary of the organism that generates a velocity field v(z, z) can be determined from the asymptotics of that velocity field. Specifically Arot ■^SW
dz
— Im d> Jlim\z\ !^oo
27r
.
z),
(7)
rz)
(8)
*
and Atr
dz , 2nizViz
-1 n = -1 = < A, Ae+1-(n + l ) ^ i | ^ A r + ^ ^ ^ ( n + l)Ar,n0 Xvo(a) = ACT AU_I (cr) = A Xv-2(cr) = ACT-1 Xvn(a) = Xan+\n< - 2 ACT n + 1
Vertical Projection 0 Arot = ImX Atr = X Arot = ImMX 0
(19)
Caveat: the results for the rotational part are not correct, see Proposion 3 in section 4. To compute the Stokes curvature we compute the Lie bracket of horizontal vector fields, re-expand the resulting vector field in terms of the Fourier basis, then use the above table to determine the corresponding rigid rotation and translation. As an example if m,n > — 1, [VVn,evm} = (er)(m + 1)CT m—n-f 1
oo
erj(n + 1)CT'n - m + l \) ]V"* >JMCTY, fc=0
(20)
39 comparing this with the above table to determine the vertical components we find that for m,n > —1
{
(m + VjM^r0
n-m=j>0 and j is odd otherwise. (21)
(n + 1)M V f%n = ptr
u
_ ptr
- n = k>0 and k is odd otherwise.
m
.= 0
(22)
(23)
For the circle M = 0 and the result of Shapere and Wilczek is recovered, the only modes that couple are those that differ by one. If M ^ 0 all modes that differ by an odd number couple, although the strongest coupling takes place between ones that differ by just one. The effects of coupling between distant modes becomes more pronounced the further the shape is from a circle. 3.1
Examples and discussion
Two examples will be presented, using only the translational part of the con nection (so that the previous calculations are correct). The first, an analog of the example of Blake's swimming circular cylinders 10 , under symmetrical deformations. The second example will be that of a long thin organism that swims utilizing the undulatory mode. The results of this model will be com pared with observations of the swimming motions of nematodes by Gray and Lissman. For the first example consider an ellipse with semi-axis 1 — M and 1 + M, and swimming stroke parameterized by: M S{a,t) = (a+ —) + (.025 cos 2irt)vu + (.025sin27rt)ui 5 (7
+(.015 cos 2wt)vls + (.015 sin 2irt)vig
(24)
To calculate the net translation of the swimmer the curvature components corresponding to the coupling of modes 14 and 15,14 and 19,15 and 18, and 18 and 19 are needed. After one swimming stroke the swimmer's net translation
40
4
5
6
Figure 1. Ellipse (M = 3) at t=0, 0.25, 0.5, and 0.75
is given by: / •7ri4j5ai4 — a0-\ 1 7T + ... • z z* * = *=i + ^ + ... (30) z z2The presence of a 0 is a "confession of debt" to Stokes' paradox, since there is no solution vanishing at infinity for the translation of a cylinder. Write
/KO
\e=i
)
with (1.2)
V
*
= MZ)
jvrkeal=^
T T T
- Z(P^Z) = 1 4
m
- Hz" ~ k*zk-X)
Thus for each k < 0 we have four terms, taking for xp and either zk or izk. This is called the canonical basis. Observe that Vk have good symmetry properties with respect to conjugation z —> z and reflection z —> —z. However, Wk do not have these symmetries. This is already one good reason to suspect that the canonical basis is not the best choice. In Appendices A and B we obtain the curvatures for the circle, using the canonical basis, in Appendix C we show that in the case of the ellipse, the canonical basis does not allow a practical computation.
44
Let z = z(£) the conformed map taking the exterior of 7 = S 1 to the exterior of C. We transport "hodographly" the Fourier basis {cr n + 1 } in 7 to C, that is: Vn(s) = ( £ ( s ) ) n + \
n€Z.
We adopt the indexing n —» n + 1 just to maintain the convention used by SW. In contradistinction with the canonical basis, the symmetry properties are preserved by the hodographed Fourier basis. Suppose, for instance that C is symmetric with respect to the x-axis. Taking real coefficients, we pro duce symmetric deformations for all n. With imaginary coefficients, we get ondulating deformations. The Fourier basis has other advantages. For any conformal map, half of the problem to find the Stokes extensions is trivial: for n > — 1 we take always ip = fn+1 and 4> = 0. As we will see in section 5, there are also remarkably simple properties for the power expenditure operator. 4-2
Total force and total torque
A simplified version of Stokes' paradox is given by the following Lemma 1 If the potentials ij) and <j> have no singularities at 00, as in (30) then the total force vanishes:
I fds = 0 Proof. This follows from the identity fds — —2nidU where
U = 4>(z) + zp-xp . In Appendix E we present a derivation of the force field / = 4fiRe(' n) - 2/i( z W - W) n . (31) Actually, Stokes' paradox implies that in order that the total force be ^ 0, the potential U must contain a logarithmic term. Following Blake 9 , Shapere and Wilczek1 avoided the logarithmic singularities altogether, using the following Criterion 1 Expand a vectorfield v(z) along C with respect to the canonical basis (without logarithmic singularities). Then the translational component of the connection 1-form is the constant term ip0.
45
For the rotational component of the connection, we now present a correction1 regarding the rotational part. Criterion 2 In dimension two the rotational part (as in dimension 3) is de termined by the condition "total torque T = 0 ". In fact, we have Lemma 2 Using the canonical basis, T = 47r/j/m[6_i] Proof. A simple calculation gives tds = Im[z/]ds = -2fj.Re[zdU] = -fidF where F = zz(& +4>')-z7p-zlp
+^ + l
e Z=
I il>{z)dz JZo
Observe that £ is multivalued if b-\ ^ 0. Remark 1 The reader may feel unconfortable with the presence of the con stant rigid translations of the fluid as a whole, which do not decay at infinity: 1 and i as allowable vectorfields. Can one do differently? the answer is yes. Translations of the shape can also be induced by stokeslets. However, these have even worse behaviour, logarithmic at infinity14. Thus, in two-dimensions, one can still define translation-horizontality us ing the criterion "total force = 0", but this requires replacing the constant vectors 1 and i by the two stokeslets, on the x and y directions. In a future work we plan to check if the criterion using stokeslets yields the same results for the connection. 4-3
The 1-form of the connection
Given a velocity field V along C, we denote (a, b) the translation and u>k the infinitesimal rotation such that V — uiz — (a, b) is horizontal. First, we introduce terminology. Definition 1 If(a,b) = (0,0) we say V is translationally-horizontal. 0 we say V is rotationally-horizontal.
Ifu> =
46
Along C, we expand V in terms of canonical components oo
4
«w=^+EE«* • k) (37)
are rotationally-horizontal. Conjecture 1 Following Ken Meyer's talk in this Conference, our bet is 95%: All the remaining elements of the hodographed Fourier basis are rotationallyhorizontal (that is, for k > 0 and for k < — (n 4-1) )■ This involves finding the asymptotics of the Stokes extensions for the corresponding elements of the hodographed Fourier basis. We now give a more intrinsic version for (36). In order to obtain Arot(v) = a we require that (v — aiz)\c produces zero total torque. De note T the operator yielding the total torque. From T(v — ctiz) — 0 we get a = T(v)/T(iz) . Theorem 2 rot _ -^Ep lim Im — & Airot(, (v)A = v{z)dz u(x :
oo \ liri Jtz\ = R * J\t\
where Tc = T(iz) is the torque associated to the infinitesimal rotation iz(s) ofC.
48 4-5
The ellipse. Rotational components.
We saw, in section 3 the expressions for the Stokes extensions 1, a_l
<M0 =--4 +
+ •••,
bki
MO --• ^
+
-
We are interested in finding which A; have nonvanishing coefficients 6iLj. We claim that only UQ A and U_2,A do have them. In fact, since ( A _
Afi
,n = 0
ifn+1 > 0, then <jn+1 is an eigenvalue ofV"1, with eigenvector — (n + 1). It is interesting to look what happens geometrically in the physical bi plane. The force field / G Vc> is parallel (with opposing sense) to the velocity field v(z(a)) = c n + 1 . The scale factor depends on the conformal mapping, ds/dd To compute the power expenditure V, the scale factor disappears when we change the integration to 7. The difficult part of the spectral problem involves only the indices n + 1 < 0. The following proposition shows that they form an invariant subspace. Teorem 5 The operator Vy(v) = g leaves invariant the subspace of nega tive Fourier modes an+1. Proof. Use Lorentz reciprocity. Let m > 0; then (V(a " + 1 ) , am) = (V(am),
an+1) = f -maman+ld9
= 0
52
5.2
Computing g for the ellipse; negative Fourier modes
Recall that for p < —1
4"-H'*',. *
= A (
P +
1 ) ^ - ^
We insert in (39-40). In the denominators appear factors 1 — Ma2, coming from the derivatives of z(£) and £ 2 — M, followed by conjugation (remember a = o-1). Well, for 0 < M < 1
n=0
We know, however, that in the final result for g, only negative powers of C do appear. The term /J
=
^
( 7
= A(p + l K + 1
is in accordance, but the terms. I, 777, IV, V are problematic. Nonetheless, a "miracle" occurs: Lemma 6 For the conformal mapping z = R(£ + M/£) of the exterior of unit circle onto the exterior of the ellipse C, let f{Xap+1) denote the vector of forces associated to the hodographed vector field Xvp+\(o-) = Xap+1, let gd6 denote the pullback of the differential form f ds to the unit circle, so f ds = gdO (see Proposition 3), then I + III + IV + V = 0, in particular for p + 1 < 0, f{Xap+i)ds
= X{p+ l ) ^
1
d6.
Proof. A direct calculation. Theorem 6 For the ellipse, the hodographed vector fields v(a) = ap+1 for p + 1 < 0 are also eigenvectors ofVy : v —> g, with eigenvalue p + 1. We think that this is indeed remarkable: when we vary the eccentricity 0 < M < 1, using the hodographed Fourier basis, the power expenditure functional does not change! 5.3
Hypotrochoids
The degree of algebraic difficulty increases, but we can still calcultate, g(Xap+1)d6 for the hypotrochoids 11
—(«♦£).
(42)
53 with m = 1,2,..., and M a positive constant satisfying 0 < mM < 1. We present the results: Theorem 7 Consider the hodographed Fourier basis vn(a) = 0 vn(a) = a n + 1 is an eigenvalue of the Lorentz operator V (see 41) with eigenvalue, — (n + 1). 2. For m = 2 and n + 1 < 0 the "miracle" (see Lemma (6)) still holds, namely if f ds = gd9 is the pullback ot the form of forces and g = I + II + III + IV, asi in (40), then I + III + IV + V = 0. 3. For m = 3 and p + 1 < 0 the same "miracle" holds, except for p = —2, here for the hodographed vector field Au_2(c) = ACT-1 ,
g(Xo-') = (-/? + Mfta-1, /? =
^ ^
so the eigenvalue is different from the others. 4- For m — 4 we found g(Xa-1) = -Xa-1+
ACT"2
Here the hodographed Fourier basis is not an orthogonal basis for the power expenditure operator. 6
Efficiency of an elliptic microswimmer
There are basically two competing efficiency notions for microswimming. Con sider a swimming stroke of period r, in shape space S and the ratio X E
X/T E/T
mean velocity mean power
.
Froude 's efficiency is an non-dimensional quantity which results from mul tiplying this ratio by a "characteristic force" T, and in the Stokesian realm it is equivalent (up to a shape dependent factor) to Lighthill's efficiency (mean velocity) .„,. — . (44) mean power The other notion, which we call SW efficiency (because Shapere and Wilczek essentially use it) is given by efz, =
etsw = —£ ,
(45)
54
where the presence r in the denominator is equivalent to consider all swimming strokes with period T = 1. In this paper we will consider the latter notion, and consider a variational problem first introduced by Shapere and Wilczek leading to the most efficient strokes. For a comparison of the two notions, see 15 . 6.1
The variational problem
We expand the time-dependent, small shape deformations in terms of a basis, so that the problem is linearized to a Lagrangian £ = ^{K4A) ~ v-(Tq,q)
.
(46)
where the matrix K encodes the power expenditure, F the curvatures, v is a Lagrange multiplier. The Euler-Lagrange equations are Kq = vTq .
(47)
In other words, C describes the problem of minimizing the energy expenditure E =\ j\Kq,q)dt
,
X = \ f
.
for a prescribed holonomy (Fq,q)dt
Inserting q = eVeint
,
(48)
, A= — v
(49)
, S = AT * .FAT * skewsymmetric,
(50)
where V is an eigenvector, gives TV = \KV Define W = fCW so that: BW = \W Taking into account that
T
=
2TT/Q,
ef-—
.
(51)
the efficiency can be written as ,
(52)
55
where we omit the factor 27r. Some simple manipulations with E and X give E = ^Q2€2(fCV, V)T X =-l-ne2(FV,V)T
.
Since TV = XK.V, we get e f = ^ - = -iA,
(53)
Recall that A = iQ./v is an eigenvalue of (51), and as expected, it is purely imaginary. In short: Finding SW's efficiency is equivalent to find the supremum of the purely imaginary spectrum. 6.2
Numerical results
For the elliptical swimmer, undergoing symmetric swimming strokes, matrices K and F are given in Appendix F. We found the spectrum using MathLab, for M = 0 (circle), M — 0.1 e M = 0.2 (ellipses with small eccentricity) and M = 0.8 e M = 0.9 (high eccentricity). Regarding the number of Fourier modes, we solved the problem exciting the first five, the first ten, the first 30, and finally the ten modes between 30 and 39. Tables of results are available upon request. Our observations are as follows. As expected, the absolute values increase as more modes are considered. Interestingly, they do not diminish significantly as we exclude the lower Fourier modes. This indicates, as shown by SW for the circle and the sphere, that the more efficient strokes involve essentially the high geometric modes. In the table below we list the highest eigenvalues (51), for small and for high eccentricities. First 5 modes First 10 modes First 30 modes 10 modes, from 30 to 39
M = 0 1.9906i 2.4008i 2.7016i
M = 0.1 1.9664i 2.2845i 2.4884i
M = 0.2 1.9431i 2.1826i 2.3139i
A* = 0.8 2.4541i 3.31621 4.91681
M = 0.9 2.7878i 4.2483i 7.2448i
2.6529i
2.4717i
2.3134i
4.0299i
5.6556i
We have also considered very high order modes, between 30000 e 30009, and we verified that there seems to exist a higher bound for all efficiencies, as conjectured by SW 2 .
56 The advantage of high order geometric modes supports Nature's choice for ciliary motion rather than deformations of a membrane. The mechanical stress would be too strong for a membrane undergoing high order deformations. On the other hand, it is quite easy to a ciliary envelope to emulate these motions. Acknowledgments Thanks to LNCC/CNPq for supporting a scientific visit of one of the authors (J.D.) to Rio de Janeiro, February 1998. Appendix A. The Lie bracket Let z = x + iy,z = x — iy. Denote d_ _ 1 fd_ _ .d_\ % dz ~ 2 \dx dy)
d__l(d_ ' dz~ 2\dx+
.d_\ dy) '
l
A vector field v = (a(x, y), b(x, y)) = a— + b— is represented as v = f
dz
+ f
a=z
where / = a(x, y) + ib(x, y) = f(z, z). Lemma 7 The Lie bracket w = [u, v\ can be computed as follows:
w = [u, v] = 9z £;
9z 9z 9z.
or as
7" 7.
fz fz Jz fz.
-
9 9 .
(54)
L d Td w = h— + h—, az oz
with h = 9zf + 9z-f ~ fz9 - h9Consider, for n, m < 0, the vectorfields vn = zn , um = zm-
razz™-1
(55)
57 written as differential operators as fi
ft
Vn=7l—+Zn az
— az
u m = tzm - mzz™-1)—- + (zm - rrizz™-1) — . az az We obtain [vn, Vp] = pznzp~1
- nzPz"-1.
(56)
Curvatures for the circle For the circle Sa : r = a, the translational component of the curvature is the constant term in the Fourier expansion of these brackets alongo Sa- For n < p < 0 we get Ftr(vn,vp)
= an+pFtr(a^,aM) 1
=
=
2
f*
^ L f
2 n
(pe(n-^Dtf _
ne — 1 we have: [A„t/„, Amum] = ^T, " ^ T ^ " 1 ~
n Mk(T n-m+1+2*; bliYJ
)
\J
k
_ J2 ^Hm
+
1)Mkan+m+1-2k
k
+ J2 ~^.(m + l)(jfc + l)Jlf
fc
= / -rig. The velocity components are u — f and v = —g. Since / and g are harmonic, Au = Av = 0, Stokes equations are satisfied with pressure p = 0. We compute the stress tensor: en =fx =
^ifx+9y)
ei2 = o ( / y - 9x) = fy = ~9x
e22 = ~9y = - g C / x + S y )
We write the normal vector as n = p + iq. Then
fenei2\fp\
\e21en)
\q)
=
Hxfy
\fy-fx)
'fx-fy\
\.(P\ \q)
[P
Jv fx J \-q, = (en + ien)(p - iq) = V n
60
Now we compute with = F + iG. The components of u + iv = <j> — z' are u = F - xFx - yGx v = G + xGx - yFx A short calculation gives ux = -xFxx
- yGxx
Uy = Fy — lFXy — GX — J/GXy vx = 2GX + xGxx - yFxx =
Vy
X(jTXy
y+Xy
using one the the Cauchy-Riemann equations, namely Fx = Gy. Using the other one, Fv = —Gx we obtain: Au = - 4 F « , Av = ~4Fxy . Hence, the pressure is P=-^Fx
= -2fi(<j>' + W),
and the components of the stress tensor are: e n = -xFxx
- yGxx
e22 = xGxy - yFxy
(en =
ei2 = 2 ("x + uy) = -xFxy
-eu)
- yFxx
In complex variables notation e n + ie\2 = —" z so that the force associated to 4> is - 2 / i i p zn + 2fi((f>' + 4>')n Everything together yields / = 2nW n - 2 # z n + 2/x(0' + ~ft)n
61 E. Matrices K and F for the elliptic swimmer We consider symmetric deformations, so all A„ = 1.
Curvatures n\m
0 0 0 1
I R
-2 0 2 0 -3 0 3
M R
-4 0 4 0
1 1 R
0 M+l R 2 R
0 0 0 2M R
2
-2
2 R
0
0
0 oM±i
1 R 2M*+M-1
1 R o M+l z R
0
0
0
2M R
0
0
2M'+M-1
n
3 R
0
0
3 R
0
0
0
0
0
0
2 R 2 o2M +M-l
R
4
0
0
0
-4
M R
0
0
0 M+l R
3
-3
R
0 1 0 0 0 0 0 0 0
1 0 2 0 0 0 0 0 0
-2 0 0 1 0 0 0 0 0
2 0 0 0 3 0 0 0 0
-3 0 0 0 0 2 0 0 0
3 0 0 0 0 0 4 0 0
2, see Albouy 2 , Cabral 6 , Easton 11 , McCord et al.20 and Smale 22 - 23 . In this work, we study the topology of a particular integrable Hamiltonian system, i.e. the synodical spatial Kepler problem which plays a main role in Celestial Mechanics. 3
Synodical spatial Kepler problem
It is well known that the two-body problem can be reduced to the Kepler problem. We can consider two kinds of referential (the inertial and the rotat ing) and two kinds of motion according to the dimension n of the configuration space, the planar when n = 2, and the spatial when n = 3. Of course each spatial motion takes place in a plane, but it continues to be a spatial motion once the phase space has dimension 3. Thus there are four different models of the Kepler problem: two with two degrees of freedom, the sidereal (synodical) planar Kepler problem, and two with three degrees of freedom, the sidereal (synodical) spatial Kepler problem.
78
These Kepler problems have been studied by several authors (see the papers [7] and [15-19] ). In this work we present the topological study of the synodical spatial Kepler problem. In fact a topological study of the four Kepler problems including the regularizations of the singularities can be found in [9] (where there are the proofs of the results here presented). By considering a coordinate system that rotates with frequency 1 around the X3-axis, the sidereal spatial Kepler problem becomes the synodical spatial Kepler problem. Thus, the synodical spatial Kepler problem is the integrable Hamiltonian system with three degrees of freedom associated to the Hamiltonian H
=
IYI2 1 2 " jx[ "
XlY2
+ X Yl
(1)
* '
with phase space {(X,Y) € (R 3 \ {(JJ.,0,0) ,(fi-1,0,0)}) independent first integrals in involution
x K 3 }, and three
C = —2H (Jacobi integral), r, lY|2 1 / -J . E = '—- ■—- (sidereal energy), M = |M| = | X A Y | (modulus of the sidereal or synodical angular
momentum).
This model of the Kepler problem is a limiting case of the circular re stricted three-body problem in rotating coordinates when the mass parameter of one of the primaries tends to zero. Thus, knowing the global flow of the synodical spatial Kepler problem we can obtain information about the flow of the circular restricted three-body problem during a finite time when the mass parameter of one of the primaries is sufficiently small. The first work on the topology of the Kepler problem was carried out by Kaplan 14 , in 1942. There he studies the foliation of the planar sidereal Kepler problem for negative values of the energy. In 1982 Llibre 15 studied the synodical planar Kepler problem describing the foliation of the phase space by //,, and the foliation of //, by Ihe, where h and c are the values of the Hamiltonian and the angular momentum, re spectively. He used Devaney's regularization 10 of the binary collision and McGehee's regularization at infinity21. In 1984 Casasayas and Llibre 7 , in their study of the anisotropic Kepler problem, considered the sidereal planar Kepler problem. They obtained the foliation of the phase space by //,, and the foliation of //, by 7/,c, where h and c are the values of the sidereal energy and the angular momentum, respectively.
79 Their study also includes the separatrix levels and the regularizations of the origin and infinity. In 1993 Llibre and Soler19 improved the foliation of the phase space by //,, and the foliation of Ik by Ikc for the synodical planar Kepler problem. In 1994 Llibre and Nunes 17 did a general study of integrable Hamiltonian systems with two degrees of freedom defined by a central force. Using the topological method, the synodical spatial Kepler problem was studied by Llibre and Martinez Alfaro16 in 1985. There the regularization of the collision singularity was considered. Subsequently in 1990 Llibre and Pinol 18 , in their study of the elliptical restricted three-body problem, included the regularizations of the collision and infinity singularities and so improved the results on the synodical spatial Kepler problem. In both papers the au thors stated the topology of the invariant sets Icm3f> i-e- they considered the independent first integrals in involution C, M$ = E + C/2 — XiY? — X2Y\ and F = M2 + Mf = M2 - Mi where M = (Mi, Af2, M 3 ). In this paper we study the synodical spatial Kepler problem taking as independent first integrals in involution C, E and M. We extend the previous results by stating the foliation of the phase space by Ic, the foliation of Ic by J ce , and the foliation of by Icem- In fact, in [9] we also give the topology of ■»m, -*c,mi -*e a n d
4
Jew
Topology of Ic, Ice and Icem
For presenting our results we need some notation. We define R + = (0, oo) and denote by Sm the m-dimensional sphere in R m + 1 of radius 1 with center at the origin 0. We note that S° is formed by two points. Throughout this paper manifold means differentiable manifold. If A and B are topological spaces the product space Ax B has the product topology. We use the symbol « to denote a diffeomorphism between two manifolds. Thus we present without proof the following diffeomorphisms: S ' x R « S2 \ {one point}, 5 1 x R 2 « S 1 x (S2 \ {one point}) 52 x R 3 « S2 x [S3 \ {one point}) S'xS'xR 2
«53\51, » S5 \ S2,
« S 1 x (R 2 s {0}) « S 1 x (S2 \ 5°) « 5 3 N { S 1 U 5 1 } ,
2
S x S x R « S2 x (R 3 \ {0}) « S2 x (S3 x 5°) » 5 5 x (S2 U S2). The results about the synodical spatial Kepler problem are stated in the following theorems. From them we obtain the foliation of the Jacobi levels Ic by the first integrals E and M. The first theorem concerns the topology of the Jacobi levels Ic.
80
Theorem 1 The Ic level of the synodical spatial Kepler problem is diffeomorphic to S5 \ S1) R i ^ x R ^ S 1 x R 4 « 5 2 X R 3 N S ' S2US1' 2 2 «S2xS2xR S US )
i/c>3, i/0 c/2. (6) If e > 0, then the orbits are in the direct space DS formed by the levels Ice « S1 x R 3 . There is a set of: S1 hyperbolic planar orbits when m = e + c/2, and S1 x S 1 hyperbolic spatial orbits when m> e + c/2. If c = 0, we have that e G [ei, 00), 0 < |e + c/2| < m < m , if e < 0, and 0 < e + c/2 < m if e > 0. Each of the Jacobi levels Ic has one component, see
85
h
e € (ei.oo) ei = e
i-ce
S1
h
SxxS3 s3
ei < e < - c / 2
S 1 xS1 x R 2
e = -c/2
3
S'xS3 s
- c / 2 < e < e2
Sl
e = e2 = e 3 3
S'xS3 s
e2 < e < 0
Sl x R J
0<e
^cem
S1 1 S xS1 l S x 5 ' x S1 51 x S 1 S2xR 1 5 x S 1 x Sl 1 SlxSl S " l l 1 S xS S " Sl x S 1 x S 1 S'xS1 S1 1 S xS1 1 5 x 51 x 51 51 x S 1 5'xR S'xRxS1
|e + c/2| < m -7713 = m = m , — 7713 = 771 < 771, —7773 < 777 < 777, —7713 < 771 = 771,
0 = m < m» 0 < m < m» 0 < m — m, 7Ti3 = m < m « 7773 < 771 < 771, 7773 < 777 = 777* 7713 = 7 7 1 = 771, 7713 = m < m , 7713 < m < m , 7713 < 771 = 771* 77I3 = 771 77I3 < 771
Table 2. T h e foliation of Ic for c = 3.
/c
e £ [ei,oo) ei = e
■"ce
S
1
S1 x 5
3
ei < e < - c / 2
S2 X
S1 xS1 x R 2
e = -c/2
R3 S1
S'x53
-c/2 < e < 0
S1 x R J
0<e
*cem
1 s l S xS1 l l S xS x 51 1 S xS' S'^xR S ' x S ' x 51 SIx51 51 x S 1 1 5 x S1 x 5 1 5'xS1 S'xR S'xRxS1
|e + c / 2 | < 771 —7713 = m = 771. —7713 = m < m » —7713 < m
< m ,
—7713 < m = m . 0 = 771 < 771, 0 < 771 < 771, 0 < m = 771, 7713 = m < m , 7713 < 771 < m , 7773 < m = 771, 7713 =
m
7713 < 771
Table 3. The foliation of Ic for 0 < c < 3.
Table 4: (1) If e = ei, then m = m, and there is one retrograde circular planar orbit
86
h
s2
eG [e ls oo) e\ = e
Ice l
s
1
3
X
S xS
X R
Sl xS'x 5'xR Sl x R J
s2
ei < e < 0 e= 0 0<e
■•cem
S1
S'xS1 S ' x S ' x 51 Sl xSl SzxRx5u 5'xRxS1 S'xR S'xRxS1
|e + c/2| < —m3 = m = —m3 = m < -7Ti3 < m < —T7i3 <m = 0= m 0< m T7l3 = m
m m» m, m» m,
7Tl3 < 771
Table 4. The foliation of Ic for c = 0.
in the level Ice « S 1 which coincides with the retrograde space RS. (2) If ex < e < 0, then the orbits are in the retrograde space RS formed by the levels Ice K S1 X S3. There is a set of: S1 elliptic planar orbits when m = - e ; Sl x S 1 elliptic spatial orbits when —e < m < m»; and S 1 circular spatial orbits when m — m,. (3) If e = 0, then the orbits are in the level I*e % Sl x S 1 x S1 x R that contains the collision or ejection orbits. There is a set of: S 2 ejectionparabolic orbits and S2 parabolic-collision orbits when m = 0, and S1 x S1 parabolic spatial orbits when m > 0. (4) If e > 0, then the orbits are in the direct space DS formed by the levels Ice ss S1 x R 3 . There is a set of: S 1 hyperbolic planar orbits when m = e + c/2, and S1 x S1 hyperbolic spatial orbits when m > e + c/2. If c < 0, we have that e € [ei, oo), 0 < |e + c/2| < m < m , if e < 0, and 0 < e + c/2 < m if e > 0. Each of the Jacobi levels Ic has one component, see Table 5: (1) If e = ei, then m = m* and there is one retrograde circular planar orbit in the level Ice ss S 1 which coincides with the retrograde space RS. (2) If ei < e < 0, then the orbits are in the retrograde space RS formed by the levels Ice ss S 1 x S3. There is a set of: S1 elliptic planar orbits when m = —e; S 1 x S 1 elliptic spatial orbits when —e<m<m.\ and S 1 circular spatial orbits when m = m«. (3) If e = 0, then the orbits are in the retrograde space RS formed by the levels Ice «s S 1 x R 3 . There is a set of: S 1 parabolic planar orbits when m = —c/2, and S 1 x S1 parabolic spatial orbits when m > - c / 2 .
87
h
he l
s 2
S1 x S 3
s
X
s2
S1 x R J
X
R
S1 x S ' x S'xl S1 x R J
e e [ei,oo) e\ = e
-*cem
1 1s S x5' ex < e < 0 S ' x S ' x 5 1 SlxSl 0 < e < -c/2 S'xR 5'xRxS1 e = -c/2 S'xRxS1' S'xRxS1 S'xR -c/2 < e 5'xRxS1
|e 4- c/2| < m —77i3 = TTI = m » - m 3 = 771 < 771, —7713 < 771 < 771. —7713 < TTl = m « —777,3 = 771 —77I3 < 771 0 = 777 0 - ( e + c/2). (5) If e = - c / 2 > 0, then the orbits are in the level Ice w S 1 x S 1 x S1 x R that contains the collision or ejection orbits. There is a set of: S2 ejectionhyperbolic orbits and S2 hyperbolic-collision orbits when m = 0, and Sl x S 1 hyperbolic spatial orbits if 0 < m. (6) If e > —c/2, then the orbits are in the direct space DS formed by the levels Ice « 5 ' x R 3 . There is a set of: S1 hyperbolic planar orbits when 777 = e + c/2, and S1 x S 1 hyperbolic spatial orbits when m > e + c/2.
References 1. R. Abraham and J.E. Marsden, Foundations of Mechanics, Benjamin, Reading, Massachussets, (1978). 2. A. Albouy, Integral manifolds of the n-body problem, Inventiones Math. 114, 463-488, (1993). 3. V.I. Arnold, Mathematical Methods of Classical Mechanics,, SpringerVerlag, (1978). 4. V.I. Arnold, V.V. Kozlov, A.I. Neishtadt, Dynamical Systems III, Ency clopaedia of mathematical sciences, Springer-Verlag, Berlin, (1978). 5. G.D. Birkhoff, Dynamical Systems, New York, (1927).
88
6. H. Cabral, On the integral manifolds of the n-body problem, Inventiones Math. 20 ,59-72, (1973). 7. J. Casasayas and J. Llibre, Qualitative analysis of the anisotropic Kepler problem, Memoirs of the Amer. Math. Soc, Vol. 52, n° 312, (1984). 8. J. Casasayas, J. Martinez Alfaro and A. Nunes, Knots and links in integrable Hamiltonian systems, J. Knot Theory Ramification 7,123-153, (1998). 9. M. P. Dantas and J. Llibre, The foliation of the phase space for the 3-dimensional Kepler problem, preprint, (1999). 10. R. Devaney, Singularities in Classical Mechanics Systems, in Ergodic The ory and Dynamical Systems I, Proceedings Special Year, Maryland 197980, A. Katok (ed.), Birkhauser, Basel, 211-333, (1981). 11. R. W. Easton, Some topology of n-body problems, J. Differential Equa tions 19, 258-269 (1975). 12. A. T. Fomenko, On typical topological properties of integrable Hamilto nian systems, Izv. Akad. Nauk (SSSR) Ser. Mat. 52 ,378-407, (1988); English Transl. in Math. USSR Izv. 32 (1989). 13. A. T. Fomenko, Differential equations and applications to problems in Physics and Mechanics, Mechanics Analysis and Geometry: 200 years after Lagrange, M. Francaviglia (editor), Elsevier Science Publishers B. V., (1991). 14. W. Kaplan, Topology of the two-body problem, American Mathematical Monthly 49, 316-326,(1942). 15. J. Llibre, On the restricted three-body problem when the mass parameter is small, Celest. Mech. 28, 83-105, (1982). 16. J. Llibre and J. Martinez Alfaro, Ejection and collision orbits of the spa tial restricted three-body problem, Celest. Mech. 35, 113-128, (1985). 17. J. Llibre and A. Nunes, Separatrix Surfaces and Invariant Manifolds of a Class of Integrable Hamiltonian Systems and Their Perturbations, Mem. of the Amer. Math. Soc, 107, n° 513, (1994). 18. J. Llibre and C. Pinol, On the elliptic restricted three-body problem, Celest. Mech. 48 , 319-345, (1990). 19. J. Llibre and J. Soler, Global flow of the rotating Kepler problem, in Hamiltonian System and Celestial Mechanics, Advanced Series in Non linear Dynamics, Vol. 4, World Scientific, Singapore, 125-140, (1993). 20. C. K. McCord, K. R. Meyer and Q. Wang, The integral manifolds of the three body problem, Mem. Amer. Mat. Soc, 132, n° 628, (1998). 21. R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differential Equations 14, 70-88, (1973).
89
22. S. Smale, Topology and mechanics, I, Inventiones Math. 10, 305-331, (1970). 23. S. Smale, Topology and mechanics, II. The planar n-body problem, In ventiones Math. 11, 45-64, (1970).
U N B O U N D E D G R O W T H OF E N E R G Y I N P E R I O D I C P E R T U R B A T I O N S OF GEODESIC FLOWS OF T H E T O R U S AMADEU DELSHAMS Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain E-mail: [email protected] RAFAEL DE LA LLAVE Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA E-mail: [email protected] TERE M. SEARA Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain E-mail: [email protected]
1
Introduction
The goal of these notes is to summarize the main ideas of a paper by the authors. l We will study the effects of a periodic external potential on the dynamics of the geodesic flow in the (2-dimensional) torus. This is a Hamiltonian system with two and a half degrees of freedom with Hamiltonian H{p,q,t) = ^gq{p,p) + U(q,t). Here g is the metric on the torus, and U is the external periodic potential. If U = 0, the system is autonomous and, therefore, the energy of the geodesic flow Ho(p,q) = \gq(p,p) is preserved. Note also that when the metric is flat, the geodesic flow is integrable. We will establish, using geometric methods, a result that had been es tablished by J. Mather using variational methods. 2 Namely, that for generic metrics and potentials—in particular for arbitrarily small potentials and for metrics arbitrarily close to integrable—, one can find orbits whose energy grows to infinity. More precisely, we will indicate the proof of: Theorem 1 For every r = 1 5 , . . . , oo, u, there exists a Cr-residual set in the set of metrics g on T 2 , and of periodic potentials U : T 2 x T 1 —♦ R, such that the Hamiltonian H(p,q,t) = ^gq(p,p) + U(q,t) has an orbit of unbounded energy.
90
91
The result is somewhat more surprising if we realize that, for high en ergy, the potential is a small perturbation of the geodesic flow and that since there are two time scales involved (see subsection 2.2), the frequency of the perturbation is smaller than the frequency of the geodesic flow. In this sense this result can be understood as an analogue to Arnold diffusion. The difference between this work and other approaches to Arnold diffusion in the literature comes from the fact that here, the unperturbed system, which is the geodesic flow, is a two degrees of freedom Hamiltonian system that, in general, is not integrable and it is assumed to have some hyperbolicity properties. Indeed, the hyperbolicity properties involve that the system contains hyperbolic sets with transversal intersection in an energy surface. This is somewhat stronger hyperbolicity that the so-called a-priori unstable systems in the notation of Chierchia and Gallavotti 3 . We propose the name a-priori chaotic for this kind of systems, where there are some conserved quantities, but there are hyperbolic orbits with transverse heteroclinic intersections in the manifolds corresponding to the conserved quantities. This result on instability has been proved by J. Mather using a variational approach (the differentiability requirements were much smaller). The proof presented here will rely on standard tools in the study of dynamical systems. Most of them have been used traditionally in Arnold diffusion. We also use very strongly the theory of hyperbolic dynamical systems. We hope that this can serve as a clarification of the relation between variational meth ods and more geometric methods. We also point out that S. Bolotin and D. Treschev have presented another geometric proof 4 . The main difference with the method presented here is that they rely more on KAM theory and they do not use the theory of normally hyperbolic manifolds. The generic conditions that we need can be described rather explicitly. We require for the metric that the geodesic flow possesses a hyperbolic peri odic orbit with a homoclinic orbit. Once we fix this periodic orbit and the homoclinic connection, we require on the potential that the Poincare function £ ( r ) computed on it is not identically constant (see (24)). The proof can be divided in several steps that are largely independent. As we mentioned before, each step only requires almost standard tools. Let us go over a thumbnail sketch of all these steps and they will be fleshed out in the following sections. The same method of proof allows to prove somewhat more general results. If we assume the consequences of one step, then all the subsequent ones remain valid. Moreover, we can use the same method to obtain more consequences.
92 Indeed, the proof will be established by the somewhat more general Theorem 18. Riemannian
Geometry
It was shown already by Morse—with some improvements by Mather—that a generic metric possesses a hyperbolic closed geodesic with a homoclinic orbit. Note that this shows that the geodesic flow is not integrable and is a-priori unstable in the sense of Chierchia and Gallavotti. 3 It is a crucial observation that, when we consider the geodesic flow as a 4-dimensional dynamical system, a closed geodesic lifts to a cylinder A (one periodic orbit for each energy surface). This cylinder is invariant and normally hyperbolic. Scaled variables As can be seen by elementary dimensional analysis, if we rescale the momenta and the time, we can transform the perturbation introduced by U for high energy as an small and slow perturbation of the geodesic flow. Intuitively, if the geodesies are moving very fast, in their time scale, the potential is changing slowly. For the analysis that will follow, the slowness is much more important than the smallness. Another elementary remark is that we can consider the system as au tonomous by considering one extra angle variable. We can come back from this extended phase space to the regular phase space by means of the Poincard return map. In particular, the cylinder A is invariant under the action of the unperturbed Poincar6 map, and it lifts to a 3-dimensional manifold A, invari ant under the unperturbed extended flow. In this sketch, we will be somewhat cavalier about distinguishing between the objects in the extended phase space and in the regular phase space. The theory of normally hyperbolic manifolds Since A is uniformly hyperbolic in the extended phase space, we can find a manifold close to it that is invariant for the perturbed system. The stable and unstable manifolds of this invariant manifold still intersect as transversally as possible. The importance of this invariant manifold is that it will serve as a template for many types of orbits.
93 Averaging theory Taking advantage of the fact that the perturbation is slow, we can study the motion on the perturbed invariant manifold using the method of averaging. Assuming that the system is differentiable enough, we can perform several steps of averaging so that the system on the 3-dimensional invariant manifold is a very small periodic perturbation of an integrable Hamiltonian system. KAM theory Once we know that a system is close to an integrable Hamiltonian system, we can find out that it is covered very densely by invariant tori, emerging from invariant circles of the Poincare map. Poincare-Melnikov theory It turns out to be possible to compute perturbatively the gain in energy for a homoclinic excursion. Under some generic hypotheses, it is possible to show that a homoclinic excursion leads to a jump from one KAM torus to another, giving rise to heteroclinic orbits between them. This idea can be formulated precisely by introducing what we call the scat tering map which relates the asymptotics in the future with the asymptotics in the past. This map can be constructed precisely using hyperbolicity theory and it can be computed perturbatively using the Poincare-Melnikov method (the fundamental theorem of calculus and a—somewhat delicate—passage to the limit). Under the assumption that the scattering map moves enough to jump over the gaps between KAM tori—which we have already shown are very small—we can construct pseudo-orbits whose energy grows unboundedly. These pseudorbits can be described as follows. When the potential is in such a condition that produces a jump of energy in the direction that we want, we execute the jump, following a heteroclinic orbit, and in the situations when it is not favorable, we just bid our time near a KAM torus. Shadowing lemma We need to construct orbits that follow these pseudo-orbits closely. This can be done by a topological argument very close to the original one invoking the obstruction property. 5 ' 6 Now, we turn to the task of fleshing out this sketch.
94 2 2.1
The m a i n s t e p s of t h e proof Riemannian geometry and dynamics of the geodesic flow
Let us begin by describing the dynamics of the geodesic flow. The phase space of the geodesic flow is T*T 2 = R 2 x T 2 . We will denote the coordinates in T 2 by q and the cotangent directions by p. The geodesic flow is a two degrees of freedom Hamiltonian, with Hamiltonian Ho(p,q) = ^gq(p,p), with respect to the canonical symplectic form Q, which, moreover, is exact: Q = dd. In local coordinates, 9 — J^Ptdgt, Cl — Y^i^Pi A dqi. We will denote by $(*;Pi9) = $t(Pi 0, tBo = UE>E0^B - [E0, 00) x T 1 x T 2 , that is, the phase space is foliated by the energy surfaces. Given an arbitrary geodesic "A" : R —» T 2 , parameterized by arc length, we will denote by Ae(t) — (XE(t), X%(t)) the orbit of the geodesic flow that lies in the energy surface E#, and whose projection over q turns along the range of "A". Moreover, we fix the origin of time in XE so that it corresponds to the origin of the parameterization in "A". (Formally Ho(XE(t)) = E, and Range("A") = Range(A|), "A"(0) = A«,(0).) Note that
(A|(*).A^(t)) = (V2EXp1/2 (yZEt) ,A«/2 (V2Et)) .
(1)
So that, in this situation, the role of E is just a rescaling of time. The following theorem is due to Morse and Mather: Theorem 2 For a Cr generic metric g,r>2, and for any value of the Hamil tonian Ho(p,q) = E > 0, there exists a periodic orbit Ag(t), as in (1), of the Hamiltonian flow associated to Ho whose range AE is a normally hyperbolic invariant manifold in the energy surface. Its stable and unstable manifolds W? ,u are 2-dimensional, and there exists a homoclinic orbit 7B( ±00.
(2)
95 We suppose that the length of the unit geodesic "A" on the metric g is 1. Note that the speed of a unit geodesic is 1 and, therefore, its energy is 1/2. Then "A" as an orbit of the geodesic flow has period 1 and energy 1/2. Because of this we use the notation Ai/2The quantity A = a + - a_ is called phase shift. In this situation the energy is preserved and this makes the periodic orbit non-hyperbolic in the whole space. In order to obtain some hyperbolic object it is natural to discuss what happens for all energy surfaces considering A = U B > E ^-B for all values of the energy. Lemma 3 Define A = U B > B &■£• This is a 2-dimensional manifold with boundary which is diffeomorphic to [EQ, CO) X T 1 , and the canonical symplectic form on T*T 2 Q restricted to A is non-degenerate and invariant under the geodesic flow $ t . The stable and unstable manifolds to A, W\, W%, are Z-dimensional man ifolds diffeomorphic to [EQ, co)xT 1 xR, and7 = U E > £ 0 1E is 0,2-dimensional manifold diffeomorphic to [£0,00) x R that verifies 7c(WX\A)n(wx\A).
Let us note that the intersection of the stable and unstable manifolds along 7 is transverse, because TxW\nTxWl
=
Txl.
This meaning of transversal intersection has several important consequences. For example, 7 will be a locally unique intersection. Since the manifold A is a cylinder, we are going to take a coordinate sys tem in A, given by one real coordinate (momentum) and one angle (position). The real coordinate will be J — \/2Ho > T/2EQ. For the angle coordinate, we will take (f € T 1 , which is determined by dJ A dtp = fi|A and ip = 0 corre sponds to the origin of the parameterization in "A". We can identify a point in A with the values (J, (p) taken by these coordinates. The coordinate repre sentation of the periodic orbit AE( £s '>
6 = e
2
q=^(p,q),
> (10)
5 = 1,
defined on the extended phase space T*T 2 x ^T 1 . The reason why we introduce the notation 6 = e2 above is so that we can study more conveniently the limit e —► 0. The normally hyperbolic pertur bation theory can make statements for the above system for all 6 < 60 and one can choose 60 independently of e. This allows us to make statements for £ small enough, even if the scaled flow is meaningless in this limit. We note that for the unperturbed case (5 = 0), we have that A := A x i f 1 ~ [Jo, 00) x T 1 x -T1 £
£
is a 3-dimensional manifold locally invariant for this flow. This manifold is also normally hyperbolic. The next theorem assures the persistence of a hyperbolic manifold for e > 0 close to this one, and it relies in results of hyperbolicity theory. x
99
Theorem 4 Assume that we have a system of equations as in (10), and of class Cr, 2 < r < oo. Then, there exists em > 0 such that for \e\ < e», there is a C r _ 1 function T: [Jo + # e 2 , o o ) x T ' x ^ T 1 x (0,e 2 ) -» T*T 2 x -T1 £
£
such that A£ = T ({Jo + Ke2,oo) x T 1 x ^ T 1 x {e 2 })
(11)
is locally invariant for the flow (10) and is £2-close to A in the Cr~2 sense. Moreover, A£ is a hyperbolic manifold. We can find a C r _ 1 function T* : [Jo + A ^ . o o ) x T 1 x -T1 x [0,oo) x [0,£2) - T*T 2 x ^ T 1 such that its (local) stable invariant manifold takes the form Ws'loc(Ae)
= F ([Jo + Ks2, oo) x T 1 x i f 1 x [0, oo) x {e 2 }^ .
(12)
Ifx = F(J, ip, s,e2) e A,, then W^s-loc(i) = . P ( { J } x {^} x {s} x [0, oo) x {e2}). Therefore W^8,loc(A£) is e2-close to W s - loc (A) in the Cr~2 sense. Analogous results hold for the unstable manifold. Let us observe that Theorem 4 only guarantees local invariance for A£. But, in a moment, we will show that this manifold has KAM 2-dimensional tori, that will provide with invariant boundaries for it. Therefore, it is possible to take A£ invariant. This remark leads to some extra uniqueness, etc. which could be useful for future developments. However, we will not use it in the present paper. 2.5
The perturbed inner map. Averaging theory
Once we have the 3-dimensional perturbed invariant manifold A£ given by (11), we can consider the flow restricted to it, and, after some symplectic changes of variables, we can write it as a Cr~2 Hamiltonian system with one and a half degrees of freedom, and 1/e periodic in the time s. More precisely, this Hamiltonian can be written in the local coordinates (J, !)■
where we have put /3 2 > 0 in order to fix ideas, A G R n , and A is an (n x n)matrix. With some scaling, we can assume 0 — 1, and our Hamiltonian written in the form H (x, y, (p, I) = h(y, I) + ef(x, y, , assumed Diophantine, consists of the frequencies of this torus: (p = u>. In view of the nondegeneracy condition (5), the neighboring tori have different frequencies. Parametrizations for the unperturbed Hamiltonian We denote % the whiskered torus of Ho having frequency vector UJ. This torus can obviously be parameterized by To-
z'o{ 0. Note that Xo(s) goes from 0 to 2TT when s goes from —oo to oo. It is clear that we can give the whisker Wo the parametrization W0:
z0(s,V5) = (xo(s),y 0 (s), ¥ ) + (xo(s)-7r)A,0),
s € R,
ioo
and this implies that that every trajectory on Wo is biasymptotic to two different trajectories on the invariant torus %. If A is an integer (a very special case) then these two trajectories on % coincide. 3
Preservation of the whiskered torus and its whiskers
The local normal form Before studying the splitting, we have to establish the surviving under per turbations of our Diophantine whiskered torus, as well as its local whiskers. Then we have to extend them to global whiskers in order to compare the sta ble and the unstable ones. The surviving of the torus and its local whiskers under a small perturbation can be ensured by means of the hyperbolic KAM theorem, a version of the KAM theorem adapted to this problem. Roughly speaking, the hyperbolic KAM theorem provides a symplectic transformation $ taking our Hamiltonian into a local normal form H = H o $ (in some domain), having a simpler expression in which the perturbed torus becomes transparent, as well as its whiskers. This kind of result follows from a convergent KAM-like iterative scheme. We are interested in a normal form defined in a whole neighborhood of our concrete torus, 1 , n according to the "Kolmogorov's approach" to KAM theory. This approach allows us to control a neighborhood of the local stable whisker, which can be ensured in this way to contain also a piece of the global stable whisker (this feature is used in section 6). On the contrary, in the "Arnold's approach" (used in other papers) the normal form only holds on a Cantor set, although a large family of surviving tori is obtained. Some more comments and references to papers following both approaches are given in a recent paper of the authors. 10 In most papers (like for instance n ) , the hyperbolic KAM theorem is dealt in terms of some local variables in a neighborhood of the torus, in such a way that the whiskers become coordinate planes. A significantly new approach was introduced by Eliasson, who rewrote the hyperbolic KAM theorem and expressed it directly in the "original variables". l This is more suitable to our
118
purpose of carrying out a global control of the whiskers in order to study their splitting (see section 6). Another key fact is the use of exact symplectic transformations to normal form in the hyperbolic KAM theorem. To recall what an exact symplectic transformation is, consider the l-form 77 = -(ydx + Idip), whose differential is the standard symplectic 2-form: dr? = dxAdy+dipAdl. Then a transformation $ is symplectic if the l-form 0 the complex set Br = {(x,y, if, I) : |x|, |y|, | / | , |Im p | < r} . For a function f(x, y, y>, I) analytic on some domain D (and continuous on its closure), we denote | / | D its supremum norm. Theorem 1 (Eliasson's theorem) Let H = HQ + HH\ as described in (12) and in the assumptions (4~6), with r > n — 1. Assume H analytic on Br (r < ro). Then for |/J| small enough, there exists an exact symplectic transformation $ = $ ( - ; / J ) : Bur —► Br (analytic with respect to (x,y, a = 0(M)> b = 1 + O(M)The most important point about this result is that, thanks to the use of the original variables x, 7/ u the local normal form H can be put in terms of the generalized pendulum P(x,y,I). By using this feature, a "global" control of the whiskers, very useful in order to compare them and study the splitting, can be carried out. 1,1D In is not hard 10 to establish the validity of theorem 1 in the singular case, with fi — ep and w = w*/y/e, for |e| small enough.
119
Paratnetrization of the perturbed torus It is clear that the normal form H given in (8) has a whiskered torus of fre quency vector ijj. We denote this torus as T, and its associated local whiskers as W^ (stable) and W,",. (unstable). The torus T has the following obvious parametrization: f:
i »
= (0,-(A,a),^a),
^ T .
This torus can be translated to a whiskered torus T of the original perturbed Hamiltonian H:
T:
* » = * ( r M),
