Physica D 36 (1989) 64-82 North-Holland, Amsterdam
A CRITERION FOR NON-EXISTENCE OF INVARLANT TOPA FOR HAMILTONIAN SYSTEMS R.S. MacKAY Nonlinear Systems Laboratory, Mathematics Institute, Universityof Warwick, Coventry CV4 7AL, UK Received 30 August 1988 Revised manuscript received 15 December 1988 Communicated by J.E. Marsden
A particularly interesting class of invariant tori for a Hamiltonian system satisfying the Legendre condition is the class of invariant Lagrangian graphs. A classical result of the calculus of variations is that for such systems every orbit on an invariant Lagrangian graph has minimal action. This provides a simple criterion for non-ex;.stence of invariant Lagrangian graphs, namely, if an orbit has a variation which decreases the action or it possesses c.onjugate points then there is no invariant Lagrangian graph containing it. As an example, the criterion is applied to the motion of a particle in the field of two waves, to prove that there are no invariant tori with 'initial velocity' between those of the two waves if the amplitudes of the waves exceed certain values.
L lntr~ucfion KAM theory shows that most invariant toil of a non-degenerate integrable Hamiltoniaat system persist for small enough Hamiltonian perturbations. This is extremely significant, implying stability, at least in the sense of large measure, for many systems. Most of the proofs are constructive, and with hard work they can be used to give realistic results about the size of perturbation allowed for persistence of tori with given winding ratio (e.g. CeUetti and Chierchia [9]). It would be Lnteresting, however, to know conditions under which a Hamiltonian system has no invariant tori of a given class in some region of phase space. This would then open up the possibility that some orbits could exp!ore large regions of phase space. The purpose of tiffs paper is to present and apply a criterion for non-existence of invariant surfaces of a certain class for Hamiltonian systems of arbitrary number of degrees of freedom satisfying the Leger~dre condition. The class of surfaces considered are the Lagrangian graphs, whose relevance will be discussed. I ~ve two versions of the criterion. They are simple to state and to use, both analytically and numerically, as I show by applying them to an example, namely, the motion of a particle in the field of two waves, the protot)q~e system studied by Escande and Doveil [11]. The criteria are based on a classical result of the calculus of variations, which I attribute to Weierstrass. They are closely related to the criterion of MacKay and Percival [18] for non-existence of invariant circles for area-preserving twist maps. The theory is presented in section 2, it is applied to the two-wave example in section 3, and the method is discussed in section 4. 0167-2789/89/$03.50 © Elsevier Science Publishers B.V. (No~h-Holland Physics Publishing Division)
R.$. MacKay/A criterionfor non-existence of invariant tori
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2. Theory 2.1.
Generalities
I develop the theory in the context of Lagrangian rather than Hamiltonian systems as this is where Weierstrass's theorem fives, but I will end up with a criterion which is equally applicable in the Hamiltonian context. Firstly I review some basic results in Lagrangian and Hamiltonian mechanics (see [2] for example). The orbits of a Lagrangian system on a manifold M are the C 1 paths q: R - , M such that for all t o < t I ¢ R and all infinitesimal variations 8q with 8q(to) = 8q(tl) = O, the first variation 8W of the action
W[ql = / t ~ L ( q ( t ) , ( l ( t ) , t ) d t to
is zero. This leads to the Euler-Lagrange equations~
d[~L(q(t),(l(t),t) ] ~L(q(t),(l(t),t) dt i)gl -i)q • The function L: TM x R -~ R is called the Lagrangian. In all that follows I suppose it to be C 2 and to satisfy the Legendrecondition: ~2L/a~2 is positive definite everywhere.
Example: Two-wavesystem (Escande and
Doveil [11]). The motion of a particle in the field of two waves in one dimension can be described by the Lagrangian
L(x,
2, t) = ½22 + Mcos2crx +
Pcos2~rk(x- t),
with x ~ N. Say a Hamiltonian system ~ = H~,, /~ = - H q , where subscripts denote partial derivatives, satisfies the
Legendre condition if ~2tt/~p2 is positive definite. As is well known, every such system can be turned into a Lagrangian system by the Legendre transform: change coordinates from p, q, t to q, ~, t by
O=
aH
q, t),
which is one-to-one ff
~2f'iV/~)p2 is positive definite (e.g. appendix
D of Avez [3]), and define
L(q, 4, t) = ( p , 4 ) - n ( p , q, t). where ( , ) denotes the standard inner product. It satisfies the Legendre condition since L,~,i= Hp~1. Conversely, every La~angian system satisfying the Legendre condition has a Ha~-rfiltonian formulation, by defining
66
R.S. MacKay / A criterionfor non-existence of invariant tori
and H ( p , q , t) - (P,~I) - L(q, (l, t). This gives a flow on the cotangent bundle T*M. The dimension of M is called the number of degrees of freedon: of the sysmm. Say a Hamiltonian system is autonomous if H t ffi O. For an autonomous system H is conserved, so one obtains a family of flows on the codimension-one surfaces T * M t t o - { z e T ' M : H ( z ) --- H0}. If H is non-antonomous, it can be made autonomous by defining the extended H ~ t o n i a n system
K((j,,
(q,
- n ( p , q,
+
This gives a flow on T*(M x R). The motion on any surface T*(M x R)x ° is the same as that of the original system, with ~-- 1. Note, however, that the Legendre condition is lost. If H is invariant under some group G of canonical transformations then the flow generated by H can be reduced to a flow on T * ( M / G ) . For example, the two-wave system has extended Hamiltonian
K ( p , e, x, ~)ffi ½p~- Mcos2~rx- Pcos2~rk(x- ~) + e,
which is invariant under gl: ~ ~ ~ + 1 / k and under g2:
x~x+l, ~'~-1. Hence it induces a flow on T*(Tz), where as coordinates on "f~ one can take (x, y) with y = k O ' - x).
2.2. Lagrangian graphs The phase space T*M (or its extension for non-autonomous systems) has a natural symplectie structure, given by ~ = ~ d p i A d q i ( + d e A dr), which is preserved by the flow. A C 1 submanifold ~ of a symplectic space of dimension 2n is stud to be Lagrangian if it has dimension gJHt~Jt ~ L n and ~0]~ = O. in ic~al coordinates such that ~ is a graph p = 7 q,q), Z is Lagrangian u:~ and only "~ ,k...~. :~~ a C 2 function S: @~ -~ @ such that f = Sq. This is no restriction ff n = 1, but is a restriction when n > 1. A Lagrangian graph for a s~np|~fic manifold T*M is a set of the form p = S q ( q ) for some C 1 function S: M ~ @. In the calculus of variations it is known as a Mayer field. Note that by this definition the graph need be ordy C °. In fact, Bangert and Salamon [5] have proved that under the Legendre condition every invafiant Lagrangian graph is actually Lipschitz. A similar result has been proved by Herman [13] for symplectic maps. For the case of the extended flow of a Harniltonian system, Lagrangian graphs are given by p = Sq(q, ~),
e-" S,(q, ~').
R.S. MacKa),/ A criterion for non-existence o] inoariant tort
67
Proposition. A Lagrangian graph iv invariant under Hamilton's equations if and only if S satisfies the Hamilton-dacobi equation S,(q, ¢) + H(S~(q, ¢),q, ¢) =constant. Proof. First prove it for S ~ C 2. At p = S¢(q, ~), e = S,(q, ¢),
d/dt (Sq(q. ~.) - p ) = S,qHp + S,, + H,= 8/8q (S,(q. ¢) + H(Sq(q. ¢). q. ¢)) d/dt(S,(q. ~)-e)=S~Ho + S,, + n,=a/a,(S,(q. ~) + n(Sq(q. ,).q. ~)) Hence the surface is invariant itf the Hamilt0n-Ja.eobi equation is satisfied. If S is only C 1, use a C 2 approximation argument. [] Lagrangian submanifolds and Lagrangian graphs play an important role in Hamilton/an mechanics. For example, all the invariant toil constructed by Adaold's proof [1] of persistence of invariant toil for Hamiltonian systems on T*(T n) are Lagrangian graphs, because they are C~-small canonical transformations of a surface p = constant. In fact there is a stronger connection.
Definition. An incommensurate linear flow is a flow on 0=C0,
T n
given by
O E T n,
f o r s o m e ~o ~ R ~ s u c h t h a t m ~ Z ~, ( m , ~0) = 0 ~ m = 0.
Theorem 1 (e.g. Salamon and Zehnder [23]). If ~" is an invariant n-torus of an autonomous Hamiltonian system of n degrees of freedom on which the flow is Cl-conjugate to an incommensurate linear flow, then oq" is Lag,ran#an. A proof is given in appendix A. Say an invadant torus for a Hamilton/an system on T*(T ~) is rotational if it is homotopic to the zero section p = 0. For systems of one degree of freedom satisfying the Legendre condition all rotational tori can be shown to be Lagrangian graphs, as follows.
Theorem 2. (i) If ~" is a rotational invariant 1-torus for an autonomous Hamilton/an system of I degree of freedom with H~, > 0 then .9" is a Lagrangian graph. /::x .Lit Tr ~a:. . . . #..#:,.....1 :.... ..:..., .~ ,..~...o a;!. T, rr2~ g~,,. eh,. ,~vt,~d,,, r4 c,f ~ ~m;]tcmi~n .~y.~tem ~,Jl~ r J . O Og J.g.JIbO.IgltK~']tltOIdL a x a v o ~ t a a a t z ~ , ~-v, vx~o .u Jt ~, a yKO x,~-.L . . . H ( p , q , t ) with Hpp > 0 and Z 2 symmetry, then ~" is a Lagrang/an graph. Proof. 0) Let ~ be the time-~ map of the flow, for some ~"> 0. Then #- is invariant under ~, ~ is area-preserving, and if ¢ is small enough ~ is a twist map (Moser [21]). Hence by a theorem of Birkhoff (e.g. Herman [12]), 3" is a graph. In one degree of freedom all oaeo~ca~:o;.,d oul~,,,~,.~ ,~ e Lagrangian. So ~" is a Lagrangian graph. (it) Choose coordinates (01, 02) on $ 2 such that 02 > 0. This is always possible since ~ = 1. Let Z~ be the surface of section K = K 0, 02 =~x. Let ~: (01, p)~(O;, p') be the first return map on Z~. It preserves the
R.S. MacKay/A criterionfor non-existenceof invariant tori
68
area dp ^ dq + de ^ d~'lr.. Since Hnp > 0, ~ is a composition of twist maps (Moser [21]), hence a tilt map. Thus by the extension of Birkhoff's theorem to tilt maps (Mather [19]), ~9"nZ , is a graph over 01. Since this is true for all a ~ ' l "1, if" is a graph over (q,,), say p = P ( q , , ) , e = K o - H ( P ( q , , ) , q , l " ) . It is automatically Lagrangian by the following lemma, since in one dimension P(q, ~)= O/~q yoqP(q', , ) d q ' . []
Lemma. If ~ is an invariant surface for K(( p, e), ( q, 1-))= H ( p , q , ~)+ e, ( p , q ) ~ T ' M , contained in K -- K o, such that p = P(q, ~) = Sq(q, ~-) locally, then Z is Lagrangian.
Proof. On ~, e = E(q, ~) = K o - H(P(q, ~), q, ~). Now a/aq ( H( P(q, ~'), q, , ) ) = Ht,P¢( q, ~') + Hq =4Pq(q,,)-P
since p = P(q, ~') is invariant. Therefore, since P = Sq, the 1-form
P(q,~')dq+E(q,,')d~ is exact, so Z is Lagrangian.
[]
Note that in the situation of part (ii) of the theorem, if A c T*(T2)Ko is a subset homeomorpFfic to T 2 x [0,1], homotopic to p =0, through which no rotational invariant toil pass, then by Birkhoff's theorem there exist orbits which pass from below A to above A and orbits which do the reverse. So non-existence of rotational invariant tori implies that at least some orbits wander quite a bit. Movix~g to systems with two or more degrees of freedom, one might ask whether all rotational invariant tori are Lagran#an graphs, if the Legendre condition is satisfied. Simple examples (e.g. Herman [13]) show, however, that rotational invariant toil need be neither Lagrang~an nor graphs. But it might still be that every Lagran#an rotational invariant torus is a graph. Indeed, this has been proved by Herman [13] for symplectic maps close enough to "complete integrable" ones. 2.3. Weierstrass ' theorem The criterion ! will present is for non-existence of invariant Lagrangian graphs. I will give two formulations. The orbits of a Lagran~an system are the paths of stationary action. Say an orbit q: R - , M has minimum action if Vt 0 < t 1 and paths ~ with ~(t0) = q(to), (l(q) = q(q), then W[q] <_ W[~]. Say it has non-degenerate minimum action if in addition, Vt o < t 1 the second variation 82W in the space of variations 8q with 8q(to)= 8q(t~)= O, is positive definite. In fact, under the Legendre condition, all orbits of m~um action have non-degenerate minimum action (e.g. combine theorems 3.1 and 5.2 of Hestenes [141.
Theorem 3 (Weierstrass). If Z is an invafiant Lagrangian graph for a Lagrangian system satisfying the Legendre condition, then every orbit on Z has non-degenerate minimum action.
R.S. MacKay/ A criterionfor non-existence of invariant tori
69
This theorem was announced by Weierstrass in lectures of 1879 (see Bolza [6] for some history) but his lectures were written up only posthumously (Weierstrass [25]). Hilbert [15] gave an elegant proof in his 23rd and last mathematical problem, entitled "Further developments of the methods of the calculus of variations". In fact it is a general problem, which I paraphrase as "to give to the calculus of variations the appreciation which it is due". Maybe the present paper can be seen as a small contribution ha this direction. It is also proved by Caratheodory [8]. For a more recent treatment see theorem 7.1 of Hestenes [14]. A proof due to Salamon is given in appendix B. Note that for geodesic flows on ]-2, Byalyi and Polterovich [7] have proved a converse to theorem 3: if an invaxiant Lagrangian torus is composed of minimals then it is a graph. Theorem 3 gives a criterion for non-existence of invariant Lagrangian graphs. Criterion 1. If q: [to, tt] --, M is an orbit segment which is not a non-degenerate minimum of W, then q is
not contained in any invariant Lagrangian graph. The simplest way to use this criterion is to evaluate the matrix representing 82W for some independent variations 8qi vanishing at the ends and see if it fails to be positive definite. It is useful, however, to give another version of this criterion. Say two times t o 4= t t ~ R are conjugate for an orbit q if there is a non-zero tangent orbit liq (i.e. orbit of the linearised equations about q) with 8 q ( t o ) = 8 q ( t t ) = 0. Jaeobi proved that minimising orbits never have conjugate points (e.g. Hestenes, theorem 3.1). Hence: Criterion 2. If an orbit q has conjugate points then it does not lie on any havariant Lagrangian graph.
This formulation has the advantage that one does not need to evaluate entirely ha the Hamiltonian framework.
52W, and that one can work
3. Application 3.1. Generalities In this section, I apply criteria I and 2 to the two-wave system L ( x , ~, t ) = ½:~2+ Mcos2~rx + P c o s 2 ~ r k ( x - t ) .
By the transformations given ha section 2, we can think of this as a Hamiltorfian system on T*(T 2) with Hamiltonian K ( p , e, x , ,r ) = 12p2 - Mcos2~rx- Pcos2,rry + e,
where y = k ( l " - x). Let p = P / M . Define the winding ratio ~0~ PR (the set of rays in ~o= lJm y ( t ) / x ( t ) t - * o0
R 2) of an orbit to be
R.S. MacKay / A criterion for non.existence o1"invariant tori
70
if the lim;t exists. By a theorem of Poincard [22] and the fact that 4 = 1, the limit exists fc,v all orbits lying on a rote~tional invariant 2-torus, and is independent of the starting point. Alternatively, define the mean velocity V to be
v . - ran x ( t ) / t , t--~ oO
which exists iff the winding ratio exists. They are related by
= k(X/v-
x).
By a suitable coordinate change one can make the system look close to integrable for p large. Hence for all (M, P, k) the system has invariant tori with V large, of. Levi [26]. A region of interest, however, is V e (0,1), corresponding to mean velocities between those of the two waves. Define the initial velocity of a torus to be the velocity when x --y -- 0. Using criterion 1, I prove by hand that for k = I, = 1 there are no rotational invariant tori with initial velocity vo e [0,1] for M > M 0 =0.223718, a root of a certain quadratic equation. For comparison, based on Greene's residue criterion I believe that there is a torus with mean velocity
V= 2-I for all IMI < Mc -- 0°027587, and that there are no tori with V ~ (0,1) for all IMI > Mc (Davison [10]). Then I use criterion 2 on a computer to obtain a set of (M, o0) such that there is no rotational invariant torus with initial velocity o0 for amplitude M, with k = ~, = 1. This gives results in close agreement with those of Greene's residue method. 3.2. Estimates by hand Without loss of generality, take M, P, k >__0. I will speeialise later to k = v = 1. Every rotational invariant torus must pass over x = y = 0. Since this is the maximum ha the potential part of the Lagrangian, I expect it to be the easiest initial condition for which to find orbits with non-minimum action. Let x be the orbit with x(0) = 0, :~(0) = v0. The second variation of the action is 82W= ft~18:~2- 4~r2[M¢os2'rrx + k2Pcos2~ry] 8x2dt,
y = k ( t - x).
The system is reversible with respect to (x, y) ~ ( - x , - y ) , t ~ - t , so x ( - t ) = - x ( t ) and it makes sense to consider symmetric time intervals [to, fi] = [ - T , ~-] and symmetric variations 8 x ( - t ) = 8x(t). I will take
Sx(t) =cos
~t
(wtu~zh v,~fishes at t = :k¢). As an exercise, one eotdd try ~x(t) = q,2 _ t 2 [nsteaC. In order to estimate 82W, we need bounds on the orbit x(t). It is given by = f ( x , t) = - 2 ¢ r [ Msin2~rx- &P sin2~ry].
71
R.S. MacKay / A criterionfor non-existence of inoariant tori So
x(t)=Oot + fo'(t-s)f(x(s),s)ds. T o obtain a first approximation, use If(x, t){ < H = 2~r(M + kP). Then Ix(t) - Oct) N o w use cos 0 > 1 - 0 2 / 2 to obtain
<~2
4'r 2
8~r4 M {Volt + --~t 2
+~
+ k4p {l - volt + .~t2
~2 + _..~ ( M l ~ , o l 2 + k 4 P i l _ v o l 2 ) i
2
<_Itt2/2.
cos2~.~dt
+(Mlool+k4Pll_ool)HI 3
+-T(~+ where fl
-
2~r~/M + k 2p and
I. = 2fo/2X" cos 2 x d x
> 0,
which are evaluated in appendix C. It is a good idea to write s = ~¢, wo = {Vo{/~ , w1 = {1 - Vo{/~ , ~ = P / M . Then
B2W ~2 - ~ ~:,,. < ,~(Wo, w~, ~, ~:, ~,) = -4~
- 1 + A(w
+
+ B(Wo +
÷
Cs',
where
A =
B=
c=
1612 ~r(1 + k2p) '
16(I+
k~)I~ ,rr3(1 + k2~,) 2' 4 I , ( 1 + k,,)~(! + k ' ~ ) ~5(1 + k:'~,) 3
F o r given s, k and I,, ~(Wo, wx, s , k, v) is quadratic in Wo, wa. If @(0,0, s, k, v) < 0 then the set of wo, wa such that @ ~ 0 is non-empty, be~ag the intersection of a~ e~p~e centered on wo = wx = - Bs/2A, with the positive quadrant. Tiffs translates in the (v o, ~ ) plane to a region bounded by a v~,,,,~ . . . . hyperbola
~2 A~({Ool ~ + k ' ~ l x - ~0{~) + ~s~({~o{ + k ' ~ i a - Oo{)~ + F o r example, k = 1, ~, = 1, s = 2.05 gives the region of fig. 1.
c ~ ' + 4---~ -
I)~_
(!>
R.S. MacKay/ A criterionfor non-existence of invariant tori
72
1.4
1.2
1.0
0.8
Vo
0.6
0.4
0.2
0.0
-0. 2
-O. t4
0
|
i
i
i
i
|
|
i
i
I
2
3
t,
5
6
7
8
9
10
f~ Fig. 1. R e , on of the ( ~ , vo) plane for which (1) shows there are no rotational invariant tori when k = v = 1, s = 2.05.
Hence there are no rotational invariant t o d with v0 ~ [0,1] for 12 > 120 = 5.8062, the root of the above equation with % = 1. Since
M=
122 4~z(1 + kZv)
'
>_ ~20 is equivalent to M >_ M 0 = 0.42697. O n e can improve these restdts by obtaining a better bound on x. This can be done by performing another P/card iteration, and using IsinO[ < [01"
Ix(t)-
Votl _
M(lvo,s+ Hs~-)+ kEP(I 1 -
=4.rr 2 (Ml%l +k2Pi1-Vol)--ff +(M+kEP) =J~3(w o + k2vw~)t 3 + $:t4~ 4, where j =
6(I + k ~ ) '
E=
l+kv 48~(1 + k~v) "
VolS+
-~
H 2~] Ts }]ds
R.S. MacKay/A criterion for non-existence of inoariant tori
73
Hence
82W
#2~. < e~2(k,p,s, wo, wl) ,IT2
452
1 + A ( w 2 + k ' , ~ t ) s ~ + D(Wo + k'.,.,.)(Wo + k~-,w.)s"
+e(Wo + k',,w,),' + e(Wo + ,',,w,)~: + C(Wo + k',,wa): + L:,
(2)
where D =
64/4 3,rr3(1 + k 2 v ) 2 '
E = 16(1+ k v ) 4 3~5(1 + k 2 ~ , ) 2 F=
'
6416
":2(1 + k r ) I 7 9"'7(1 -t- ~ 2 ¢ ) 4 '
L=
4&(1 + k~) 2
9~r9(1 + k2p) a" F o r given k, ~,, s, the set of (o 0, I2) such t h a t
(~2-~<0, iS a g a i n
b o u n d e d by a piecewise h y p e r b o l a , e.g. fig. 2
/
1.4
/
/
1.2
l//¸
1.0
0.8
0.6
O.t,
0.2
0.0
\\
-0.2 t 0.4 0
;
9
5
4
b
~
?
Fig. 2. Region of the (~2, Vo) plane for which (2) shows there are no rotational invadant toil when k = ~,= 1, s = 2.1, also showing bounds on the region where all tori of mean velocity V ~ (0,1) must lie.
R.S. MacKay/ A criterion for non.existence of invariant tori
74
for k = v = 1, s = 2.1. One finds that there are no rotational [nvariant tori with v o ~ [0,1] for all > 121 --- 4.202867, i.e. M _> M1 --- 0.223718.
3.3.
N~mericalimplementation
Criterion 2 is particularly well adapted to numerical work. Let x be the orbit with x(O) = O, 2(0) = vo. Rather than looking for the first time conjugate to O, ~ will look for the first :r such that 4-• are conjugate.
0.6
f 0.4
O2
0.0
0.4
02
0,6 X
Conjugate point method: h = 0.010 gr = 50 mp= D,IO0
0 0
!
2
Conjugate point method: h
3
4
5
= 0.050 gr = 500 m p = 0.030
Fig. 3. Some orbits through (0,0) ha the (x, y ) p!a.~e, up to the first time ~" such that _+r are conjugate, for k = v = 1 and (a) M = 0.1 a n d (b) M = 0.03.
R.S. MacKay / A criterion for non-existence of invariant tori
75
>', 0.5
•=- 0.4
0.3-
02-
0.1.
0.0 0.000
0.005
0.010
0.015
0.020
0.0:~5
0.030 M and P
Conjugate point method: grid = 150 h = O.lO tmax = 150
Fig. 4. Points (in black) in the (M, Oo) plane such that the orbit with x(0) = 0 and initial velocity oo has conjugate points within time tma x = 1 5 0 .
Lemma. If 8x is a tangent orbit to a symmetric orbit x ( x ( 0 ) = 0 ) , with 8 2 ( 0 ) = 0 , gx(0)= 1 and 8 x ( ~ ) = 0, then + • are conjugate.
Proof. By reversibility 8X2(t ) = --SX(--t) is also a tangent orbit. By linearity so is 8x3(t ) = ~x(-t). B~;, 8x 3 has the same initial conditions as 8x at t = 0. Hence 8x 3 = gx, so 8x(-~-) = 0.
N
Figs. 3(a, b) show some orbits in the (x, y ) plane up to the first ~- such that i T are conjugate, for two values of M, with k = ~, = 1. Fig. 4 shows in black the set of (M, v0) from a 150 x 150 grid for which a pair of conjugate points was found with ~- < t=~, = 150. Fig. 4 agrees well with the following results on breakup of toil found by Greene's residue method (Davison [10]):
= ~' =
2
1+37 ~0 = 1 + 2~,'
'
Mc = 0.027587. M c = 0.02724.
4. E i s c ~ s i o n
4.1. Summary I have presented two related criteria for non-existence of invariant Lagrangian graphs for a class of Hamiltonian systems. The one involves computing second variations of action, the other invoh, es finding
R.S. MacKay / A criterion for non-existence of invari ,,,t tori
76
conjugate points. I have shown that it is quite easy to use these criteria to get not too bad results by hand, and very close to optimal results by computer. I used the first criterion when working by hand because it was easier to obtain estimates of 82W than of tangent orbits, and the second criterion on the computer because to get good results the first criterion would require computing a large matrix of second variations. 4.2. Special cases There are a number of special situations which are worth mentioning. 1) Area-preserving twist maps. Moser [21] showed that for every area-preserving twist map ~ there is a time-periodic Lagrangian system with L## positive definite, for which q~ is the time-1 map. Hence the results of the present paper apply to area-preserving twist maps. In fact, criteria 1 and 2 correspond to the minimum action criterion and the cone-crossing criterion, respectively, of MacKay and Percival [18]. 2) Geodesic flows. The orbits of a geodesic flow are the paths of stationary length for a given Riemannian metric g. It is simple to show that if parameterised by arc length, they are the same as the orbits of the Lagrangian L = gij(q)fli~lj, which satisfies the Legendre condition, and that nmfimum length and minimum action are equivalent (Milnor [20], s,~ction 12). In the case n = 2, Bangert [4] gave a simple proof that every orbit on a rotational invariant torus for a geodesic flow on T*(i "2) is minimising. 3) Systems of the form kinetic plus potential. For systems of the form H(p, q) = T~j(q)pipj + V(q) with T symmetric and positive definite, by Maupertius' principle (e.g. Arnold [2]), the orbits on an energy surface H = h are the geodesics of the Jacobi metric ds 2= (h - V(q))Gij(q)dq, dqj on { q: V(q) _
W[q]= where p(t) is determined from q(t) by ~ = Hp(p,q,t). So
= f p 4- Hq(p,q,t)dt. Then
where 8p is determined from 8q by 8c~i= Hp,pj8Pi + Hp,qj~qj.
R.S. MarKay / A criterion for non-existence of invariant tori
77
5) Autonomous systems monotone in one momentum variable. Suppose there is a coordinate system q = (T, Q), p = (K, P), T, K ~ R, such that H is monotone in K. Then the motion on the energy surface H = h is equivalent to the system with ttamiltonian K(P, Q, T; h) determined by solving
H((K,P),(T,Q))=h for K, with time parameterisation determined by
dT/dt= HK(( K( P, Q, T; h), P),(T, Q)) (Arnol'd, [2]). Besides reducing the number of degrees of freedom by one, this reduction has the added advantage that it is not necessary for Hpp to be positive definite. This could be useful for finding when the invariant tort of the rotating two-body problem are lost on perturbation by a third body, since in Delaunay vaciables the two-body Hamiltonian is H(L, G, l, g ) = L - 2 + G. To apply ~he non-existence criterion it suffices that Kj,~, be positive definite. The action of a path Q(T) is
W[Q] =
f aQ- K(e, Q, r; h),tr,
with P(T) determined by Qr = Kp(P, Q, T; h) = -Hp/H K. Note that
~tkp p --
K
HK
4.3. Problemsfor the future There are a number of further things it would be good to attempt: (i) Try to capture by hand the regions of non-existence in the primary resonances of fig. 4. (ii) Try to find M 0 such that for M > M 0 there are no rotational invafiant torJ for the two-wave system with mean velocity V ~ (0,1), instead of initial velocity (see appendix D). (iii) Try the method on systems wkh two or more degrees of freedom, e.g. double pendulum, N coupled rntnr¢
Tn
A o t ~ t - t t-c~n~lta~t~ ~ n l n t e #~lr~ ~ K ~ i e
£1a(i) ft~r t h o ~ n ~ r ~ n f t n n o o n t n r K ~ t ~ ~ t h
~a(O(~
= ¢~ ( ~ h o
vertical space). Integrate forward until det [ 8qy~(t)] = 0. Then there is a linear combination of ~q ~ which vanishes at t. Hence t is conjugate to 0. (iv) Try the method on systems where the invariant surfaces of interest are non-rotational, e.g. IGrkwood gaps, tori around the moon in the 3-body problem, by looking for a local coordinate system in which they would become rotational and the Legendre condition is satisfied. To use criterion 2 on z would not need to know the transformation explicitly. All calculations could be done in the original coordinates, because it w ld suffice to be able to detect when a tangent is vertical in the new coordinates.
I&S. MacKay / A criterionfor non-existence of invariant tori
78
(v) Investigate whether all Lagrangian rotational invariant tori are graphs. (vi) Implement the method on a computer with differential equation s61vers which produce rigorous bounds on the solutions, to obtain good rigorous results. (vii) What about systems for which the Legendre condition is not satisfied? (viii) It is not true that every minimal lies on an invariant Lagrangian graph (e.g. minimising periodic orbits). But for area-preserving maps, MacKay and Percival [18] gave a supplementary criterion for non-existence, which Stark [24] subsequently showed is necessary and sufficient. Can one do the same for continuous time and arbitrary dimensions? 4.4.
Symplectic twist maps
One might ask whether the method presented here has an analogue for discrete time. In fact it does. Herman [13] has proved for symplectic maps of "r ~ x R ~ satisfying a certain twist condition that every orbit on an invariant Lagrangi~ graph has minimum action. This can be used as a criterion for non-existence of invadan~ Lagrangian graphs (MacKay et al. [17]).
4.5. Conciuston
I close with a quote from Jacobi [16], which Robert HeUeman pointed out to me. Jacobi had just described a result about conjugate points, namely, that if the number of paths of stationary action fron~ a given point q(0) on an orbit to the point q(t) changes at t = ~r then 0 and 1" are conjugate points. He writes, "this theorem.., is of absolutely no importance to Mechanics." In contrast, I think the present paper sho,,vs that conj~gate points are highly relevant to Mechanics.
Acknowledgements I arn grateful to Jiirgen Moser for telling me about Weierstrass's theorem during the joint US/CERN accc!erator school in February 1985, to Dietmar Salamon for teaching me the proofs of theorems 1 and 3, to Jaume LlJbre, Lluis Alseda and the Centre de Recerca Matemafica of the Universitat Autonoma of Barcelona for their hospitality and interest during September 1987, when I began to work this out, to Cath ~rine Wattebot for her computer programming assistance, te A~so~ Solman for ~yping the manuscript, and to SERC for financial support of the Nonlinear Systems Laboratory.
Appendix A: l~roof of theorem I Let (P, Q): ~'" ~ T*M be the C 1 map conjugating the linear flow ~=~o to the flow on 5". The torus is Lagrangian if and only if L:(O)= ~kPko,Qko,- PkojQko,= O, Vi, j, V~ ~ T ~, which is the symplecfic form
R.S. MacKay / A criterion for non-existence r invariant tori
79
evaluated on tangent vectors ~/~0 i, O/OOj. Now f is continuous and is conserved following the flow. Since is incommensurate, the orbits are dense on "UTM so f is constant. If (P, Q) are C ~- then
f Ee.o,e.o,e.o,e.o,-- J Ee o,e.o, ÷ e.o,o,e. k k
(by parts)
) -0 0j
Hence f = 0 . If (P, Q) are only C t then approximate them by C z functions in the C 1 ~toFIn, to deduce the same result, m
Appendix B: Proof of theorem 3 Let ,S be an invariant Lagrangian graph p = Sq(0, r), e=S,(q,r) and write ~=~k(q,r) on Z. The function S satisfies the Hamilton-Jacobi equation
&(q, ~) + H(Sq(q, ~),q, r) --C and without loss of generality (by subtracting C from H), we take C = 0. From the relation
L(q,(1, t) -- ( p , # ) - H ( p , q , : ) and the Hamilton-Jacobi equation, we ~ce that
L(q, ¢/(q, "r), ~') = Sq(q, ~.)~b(q, ,r) + So(q, ,r) = DS, where D is the time-derivative follov.fing the flow. Hence
WIq] = f L ( q , g ,
~ ) d r = S(q~, ~1) - S(qo, ~o)
along orbits q on ~ from (qo, ~0) to (ql, ~'0Now consider the integral along a perturbed path ~ wkh ~(~r,)= q(~'i), i - 0 , 1 . Expand L(~, ~, ~') to • second order m e = ~ - , b , ~ ,.tr ) :
for some ra(~, z), ushr.g Lo = p = Sq. We can write
wiq]- j s.(~, ~)¢- s,(~: ~):~
80
R.S. M a c K a y / A criterion for non-existence of invariant tori
since this also comes to S(ql, ~'1)- S(qo, %)- Hence the difference in action
W[~]- ~[ql= fL(g;,{;,,)-L(q,4,,)dT
-- Sq(~, q') ~ - S,r (~, 'T) dT
= fL(~, ~(~, ,), ,) - G(~, ")¢(~, ") - s,(a, ,) + Loet(t~, ~, ,)cad, - fLoo(~,
'r) eed'r >_0
n,
by the Legendre condition. Furthermore, the second variation 82W[gq] = f dt LO,Oj(q, •(q, t), t) 84, 80i is positive definite.
Appendix C
Let I. = 2 f~'/2x" cos 2 x dx, "0
n > O.
Then for n > 1, 1
I.= ~'~/2""[1 + cos2x] dx = . ,
_
fo'~/2nxn_l ~sin2x2 d x
(by parts).
For n _>2, + [nx._ 1 cos2x ]~/2
.......4
n+l 1
-n+l n+ 1
[nvl "+1
~1
- 4n [['IT 2 In-'
--4 ~ n(n-
4
o
[~/2
- Jo
n ( n - 1 ) x "-2
cos2x 4
rg(n4 1~ fo~/2xn_2(2COS 2 X "
+-4 ~ ] 1)
I._ 2"
-
4
I._2
1) -
-
(by parts)
R.S. MacKay/ A criterionfor non-existence of invariant tort
fit
Io =
~-
1"1 =
'rr 2 8
2
81
-- 1.570963, 1
--- 0.7337005,
'11"3
'IT
I2 = 24
4
~r4 /3 = 64
3¢r 2 3 16 + 4" -0.4214661,
-----0.506530,
I 4 --- 0.3930328, I 5 --- 0.3962863, / 6 = 0.4231164, I 7 = 0.4720647, I s --- 0.5453575.
Appendix D: Estimates on mean velocity I would have liked to obtain a result of non-existence of rotational invafiant toil for the two-wave system for all mean velocities V ~ (0,1) rather than initial velocities %. One might hope to achieve this by estimating v0 for periodic orbits of mean velocities 0 and 1. In this appen~,c I obtain some ~uch estimates, but they are insufficient to give a non-erdstence result of the desired form. By reversibility, if x is an orbit with x(0) = 0 and x ( 1 / 2 k ) = O , if, on x is periodic w k h x(t + l / k ) = x(t), y(t + 1 / k ) = y ( t ) + 1, V=O. N o w I x ( t ) - v o t I < ~t2/2 so [ x ( 1 / 2 k ) - % / 2 k ] < H / S k 2. By the intermediate value theorem and continuity of x(1/2k) in %, there is a periodic orbit t h r o u ~ x(9) = 0 with mean velocity O, and vo > - H / 4 k . Similarly there is a periodic orbit through x(O)= O, with mean velocity 1, and vo _< 1 + H/4.
Lemma. ]if Hpp > 0 and 5 is a rotationad invafian~ toms wkb. mean velocity V and with ve]ocky vo a: x = y ~ 0 and Q is an orbit with velocity wo > vo (resp < %) at x = y ~ O, the~ Q has mean velocity ~,.,:"> V (resp < V). Proof. Suppose wo > %. The other case is similar. ~q" separates the phase space, so the whole of Q lies a b o v e ~ . N o w 4 = h ~ . W f i t e ~ e ( q , t ) f o r d l o n , q ' . T h e n H n p > O ~ > ~ , ( q , ¢ ) o n Q . HenceW>V. N
82
R.$. MacKay/A criterion for non-existence of i,~va,~ant tori
So to exhaust V ¢ (0,1) it suffices to examine the interval [ - H / 4 k , 1 + H / 4 ] in v o. The bounding lines corresponding to this interval are shown in fig. 2 for the case k = 1, = 1. Unfortunately, we do not obtain any result from this about invariant toil with mean velocity in (0,1). One can improve the estimates on vo for the periodic orbits of mean velocities V = 0,1, using the improved bound on x, but they are still not good enought to obtain a result. So further work on vo for the periodic orbits or on the region of non-existence of toil in v0 is required to achieve a result on non-existence of toil with given mean velocities.
References [1] V.l. Amol'd, Proof of a theorem of A.N. Kolmogorov on the invariance of quasiperiodic motions undei small perturbations of the Hamilton/an, Russ. Math. Surveys 18:5 (1963) 9-36. [2] V.I. Amol'd, Mathematical Methods of Classical Mechanics (Springer, Berlin, 1978). [3] A. Avez, Differential Calculus (Wiley, New York, 1986). [4] ~ . Bangert, Mather sets for twist maps and geodesics on toil, Dynamics Reported 1 (1987). [5] V. Bangert and D. Salamon, in preparation. [6] O. Boiza, Lectures on the Calculus of Variations (1904) (Chelsea, Bronx, 1973). [7] M.L. Byalyi and L.V. Polterovich, 1986, Geodesic flows on the two-dimensional torus and phase transitions "commensurability-noncommensurability", Fn. Ana. Appl. 20 (1986) 260-266. [8] C. Caratheodory, Calculus of Variations and Partial Differential Equations of the First order, Part II (1935) (Holden-Day, San Francisco, 1967). [9] A. Celletti and L. Chierchia, Construction of analytic KAM surfaces and effective stabili.ty bounds, Commun. Math. Phys. 118 (1988) 119-161. [10] C. Davison, Numerical calculation of the threshold for unlimited energy transfer for a particle in the field of two waves, Third year Applied Mathematics project, University of Warwick (1988). [11] D.F. Escande and F. Doveil, Renormalisation method for computing the threshold of the large scale stochastic instability in two degree of freedom Hamilton/an systems, J. Stat. Phys. 26 (1981) 257-284. [12] M.R. Herman, Sur les courbes invariantes par les diff~'omorphismes de l'anne~u, vol. 1, Asterisque 103-104 (1983). [13] M.R. Herman, Existence et non e~tence de ~ores invariants par des diffeomorphismes symplectiques, preprint, F~.de Polytechn/que (1988). [14] M.R. Hestenes, Calculus of Variati~m and Op6raal Control Theory (Wiley, New York, 1966). [15] D. HJlbert, Mathematical problems, Ball. ~ . Math. Soc. 8 (1902) 437-479. [16] C.GJ. Jacobi, Vorlesungen iiber Dynamik, A. Clebseh, ed. (geimar, Berlin 1866), p. 48. [17] R.S. MacKay, LD. Meiss and J. Stark, Converse KAM theory for symplectic twist maps, preprint, Warwick (1989). [18] ILS. MacKay and I.C. Percival, Converse KAM: theory and practice, Commun. Math. Phys. 98 (1985) 469-512. [19] J.N. Mather, Non-exismace of invariant circles, Erg. Th. Dyn. Sys. 4 (1984) 301-309. [20] J. 1~filnor, Morse Theory (Princeton Un/versity Press, Princeton, NJ, 1963). [21] L Moser, Monotone ,'wist mappings and the calculus of variations, Erg. Th. Dya. Sys. 6 (1986) 401-413. [22] H. Poincar~, Sur les courbes dffmies par les equations dilf6rentielles, L Math. (4) 1 (1885) 167-244. [23] D. Salmon an~i E. Zehnder, KAM theory in configuration space, Comment. Math. Heir. 64 (1989) 84-132. [24] L Stark, An exhaustive criterion for the non-existence of invariant circles for area-preserving twist maps, Commun. Math. Phys. 117 (1988) 177-189. [25] IC Weie~zLrass, 1927, Mathematische Werke, rot. 7 (Olms & Johnson, 1967), pp. 210-229. [26] M. Levi, KAM theory for particles in periodic potentials, preprint, Boston (1988).