On Birkhoff's conjecture about convex billiards
Abstract
Birkhoff conjectured that the elliptic billiard was the only integrable convex billiard. Here we prove a local version of this conjecture: any non-trivial symmetric entire perturbation of an elliptic billiard is non-integrable.
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Proceedings of 2nd Catalan Days on Applied Mathematics c 1995 Presses Universitaires de Perpignan ON BIRKHOFF'S CONJECTURE ABOUT CONVEX BILLIARDS A. Delshams and R. Ramrez-Ros Universitat Politecnica de Catalunya, Spain Abstract. Birkho conjectured that the elliptic billiard was the only integrable convex billiard. Here we provea lo cal version of this conjecture: any non-trivial symmetric entire p erturbation of an elliptic billiard is non-integrable. 1. Intro duction Let us consider the problem of the \convex billiard table": let C be an (analytic) closed convex curve of the plane R 2 , parameterized by : T ;! C , where T = R = 2 Z , in sucha waythat C is traveled counterclo ckwise. Supp ose that a material p ointmoves inside C and collides with C according to the law \the angle of incidence is equal to the angle of reection". Following Birkho 1], this discrete dynamical system can b e mo deled by an (analytic) a.p.m. in an annulus. More precisely, consider the annulus A = f z = ( ' v ) 2 T R : j v j < j _ ( ' ) jg , where the co ordinate ' is the parameter on C and v = j _ ( ' ) j cos # ,with # 2 (0 ) the angle of incidence-reection of the material p oint. In this way,weobtain a map T : A ;! A given by( ' v ) 7;! ( V ) that mo dels the billiard (see Figure 1). The function S : f ( ' ) 2 T 2 : ' 6 = g;! R dened by S ( ' ) = j ( ' ) ; () j is a generating function of T ,i.e.: @ S @' ( ' ) = h ( ' ) ; () _ ( ' ) i j ( ' ) ; () j = ;j _ ( ' ) j cos # = ; v @ S @ ( ' ) = h () ; ( ' ) _ () i j ( ' ) ; () j = j _ () j cos = V and consequently T is an a.p.m. (det DT ( ' v )= 1), and ( ' v ) are canonical conjugated co ordinates.
86 A. Delshams and R. Ramrez-Ros ( ' ) # () # Figure 1: T ( ' v )= ( V ), where v = j _ ( ' ) j cos # and V = j _ () j cos . Remark 1.1 The orbits of T (biinnite sequences ( z n ) n 2 Z A suchthat T ( z n )= z n +1 ) are in one-to-one corresp ondence with the critical p oints of the formal series (called the action ) W (( ' n ) n 2 Z )= X n 2 Z S ( ' n ' n +1 ) since these critical p oints ( ' n ) n 2 Z T satisfy the equations @ 1 S ( ' n ' n +1 )+ @ 2 S ( ' n ; 1 ' n )=0 8 n 2 Z : (1) Thus, having a generating function allows us to work with only half of the co ordinates (the base co ordinates, i.e., the ' 's). The ber co ordinates (i.e., the v 's) are sup eruous. Remark 1.2 Let C and C 0 two closed convex curves such that one is the image of the other by a similarity. Then the twoassociated a.p.m. T , T 0 ,have an equivalent dynamics since the angle of incidence-reection remains unchanged by the similarity. The map T has no xed p oints but is geometrically clear that it has p erio dic orbits of p erio d 2, corresp onding to opp osite p oints with the maximum and \minimum" distance b etween them. In these orbits the angle of incidence-reection is = 2andthus v =0.
On Birkho 's conjectureabout convex bil liards 87 To study the dynamics of these 2-p erio dic p oints of T ,itis b etter to consider them as xed p oints of T 2 , and study T 2 . But since it is not easy to nd the generating function for T 2 ,we instead intro duce a simplication. Supp ose now that C is symmetric with regard to a p oint. By the previous remark 1.2, we can assume that this p oint is the origin. Then it is p ossible to cho ose a parameterization of C such that ( ' + )= ; ( ' ) and the 2-p erio dic orbits are of the form ( ' 0 0), ( ' 0 + 0), that is, two opp osite p oints over C . Remark 1.3 We can consider the variable ' dened mo dulus in the symmetric case. Let now R : A ;! A b e the involution R ( ' v )=( ' + v ) ( R 2 is the identity). By the symmetry of C , T and R commute and it is a commonplace to use this symmetry to convert the 2- periodic points into xed p oints. Concretely,we dene a new map F : A ;! A by F = R T . Since F 2 = T 2 , the dynamics of F and T are equivalent. Moreover, since ( + )= ; (), it is easy to checkthat L ( ' ) = S ( ' + )= j ( ' )+ () j (2) is a generating function for F , and consequently F is an a.p.m. 2. Elliptic Billiards The simplest examples of convex curves are the ellipses. It is clear that the case of a circumference is very degenerated for a billiard, since it consists only of 2-p erio dic orbits. So, let us consider nowa non-circular ellipse: C 0 = ( x 2 a 2 + y 2 b 2 =1 ) = f 0 ( ' )=( a cos ' b sin ' ): ' 2 T g with a 2 6 = b 2 . Without loss of generalitywe can assume that a 2 ; b 2 =1 (wechange the ellipse using a similarity, if necessary). Thus a> 1, b> 0 and the fo ci of the ellipse are ( 1 0). Let us denote T 0 : A ;! A the analytic a.p.m. asso ciated to the ellipse C 0 ,and F 0 = R T 0 .
88 A. Delshams and R. Ramrez-Ros It is clear that the p oints (0 0) and ( 0) form a 2-p erio dic orbit for T 0 that corresp onds to the vertexes ( a 0) of the ellipse, and hence z 1 := (0 0) is a xed p ointfor F 0 .(( = 2 0) is another xed pointfor F 0 .) Birkho was the rst one in noticing that this system is integrable , i.e., there exists an analytic function H ( ' v ), called rst integral , that preserves the orbits of T 0 : H T 0 = H . As a consequence, the curves f H = constant g are invariant under T 0 .Obviously, H is also a rst integral of F 0 = R T 0 . In fact, it is not dicult to checkthat H ( ' v ) = (sin 2 ' ; v 2 ) = 2 is such a rst integral, under the assumption a 2 ; b 2 = 1. (See, for instance, 2] for the derivation of I = ; 2 H +1.) In addition to the involution R , the ellipse C 0 has another symmetry R : A ;! A given by R ( ' v )= ( ; ' v ). R is also an involution, and moreover F ; 1 0 = R F 0 R ,i.e., F 0 is R -reversible. The dynamics of F 0 , based in the level curves of H is drawn in Figure 2 where the resemblance with the phase p ortrait of a p endulum shows up clearly. 0 = 2 ; + ; ; v ' Figure 2: Phase p ortrait of F 0 ; are the separatrices of F 0 . The main prop erties of F 0 are listed in the following Lemma, whose pro of can b e found in 3]. Lemma 2.1 a) z 1 =(0 0) is a sadd le xedpoint of F 0 and Spec DF 0 ( z 1 )] = f ; 1 g , with =( a + 1)( a ; 1) ; 1 > 1 .
On Birkho 's conjectureabout convex bil liards 89 Moreover, if h := ln the fol lowing expressions hold a = coth( h= 2) b = cosech( h= 2) : (3) b) ; = f ( ' sin ' ): 0 <'< g are the separatrices of F 0 :if z 2 ; , then F n 0 ( z ) ;! z 1 when n !1 . c) If ( t )=( ' ( t ) v ( t )) , where ' ( t ) = arccos (tanh t )= arcsin(sech t ) and v ( t ) = sech t , then F 0 ( ( t )) = ( t + h ) . In other words, are natural parameterizations of ; (with regardto F 0 ). d) Let ( t )= ' ( t + h ) . Then b sin ' ( t ) + sin ( t ) j 0 ( ' ( t )) + 0 (( t )) j =sech( t + h= 2) : (4) 2.1. Entire elliptic billiard p erturbations Birkho conjectured that the elliptic billiard was the only integrable convex billiard. Our goal is to see that this is lo cally true for the symmetric billiards, i.e., any non-trivial symmetric entire p erturbation is non-integrable. (Non-trivial p erturbation means not reducible to an ellipse.) Let f C " g b e an arbitrary family of p erturbations of the ellipse C 0 , consisting of analytic curves dep ending on a C 2 wayon " and symmetric with regard to a p oint O " . Let us denote by Q " the two furthest (and opp osite) p oints over C " with Q 0 =( a 0). Using a similarity that takes O " and Q " to (0 0) and ( a 0) resp ectively, the initial family can b e put in the following form C 0 " = ( ( x y ) 2 R 2 : x 2 a 2 + y 2 b 2 + "P ( x y " )= 1 ) (5) where: I) P is analytic in x , y and at least C 1 in " , II) P ( x y " )= P ( ; x ; y " ), III) P ( a 0 " )= @ y P ( a 0 " )= 0,
90 A. Delshams and R. Ramrez-Ros or equivalently,like C 0 " = f ( ' " )=( a cos ' sin ' b + " ( ' " )]): ' 2 T g (6) where: i) is analytic in ' and at least C 1 in " , ii) is -p erio dic in ' . From I I I), it follows that P ( a cos ' b sin ' " )= p ( a cos ' b sin ' " )sin 2 ' with p satisfying also I), I I). It is easy to check that, in rst order in " , the relation b etween and P is given by P ( a cos ' b sin ' 0) = ; 2 b ( ' 0) sin 2 ': (7) Thus, if P ( 0) is an entire function in the variables x and y ,the same happ ens to 1 := ( 0). It is clear that if 1 =constant, C 0 " is, in rst order, a family of ellipses. Denition 2.1 Let f C " g beaperturbation of the el lipse C 0 .We say that f C " g is a non-trivial symmetricentire p erturbation of the el lipse when it can be put, using similarities, in the form (6) and moreover, 1 := ( 0) is a non-constant entire function. Let T " b e the map in the annulus asso ciated to the billiard in C " , where the p erturbation considered is a symmetricentire one and let F " = R T " .If " is small enough, C " is an analytic convex closed curve, and thus f F " g j " j 1 is a family of analytic a.p.m. with generating function L " ( ' " )= j ( ' " )+ ( " ) j that can b e written as L " ( ' " )= L 0 ( ' ) + " L 1 ( ' ) + O ( " 2 ), where L 0 ( ' ) = j 0 ( ' " )+ 0 ( " ) j L 1 ( ' ) = b sin ' +sin j 0 ( ' )+ 0 () j sin ' 1 ( ' )+sin 1 ()] : (8) For " small enough, the origin z 1 is again a hyp erb olic p oint of F " . This p oint z 1 lies in the intersection of the invariantcurves W s " , W u " ,suchthatany( z n ) n 2 Z orbit in the manifold W s u " tends to
On Birkho 's conjectureabout convex bil liards 91 z 1 at an exp onential rate as n ! + 1 ;1 . These invariantcurves will not longer coincide for " 6 = 0. But for " small enough, there exist op en sets U s u in W s u " and analytic functions # s u " : I s u ! R such that U s u = Graph (# s u " ), with I := I s \I u of size a little bit less than . Let us see that the primitives s u " of the functions # s u " have a nice interpretation, crucial to the derivation of the Melnikov p otential. Prop osition 2.1 Let s u " : W s u " ! R be the analytic functions denedby s " ( z s )= ; X n 0 L " ( ' s n ' s n +1 ) u " ( z u )= X n< 0 L " ( ' u n ' u n +1 ) (9) where, if z s u 2W s u " , z s u n =( ' s u n v s u n )= F n " ( z s u ) for n 2 Z .Then: 1. s u " ( z 1 )=0 . 2. s u " ( ' )= s u " ( z s u ) 8 ' 2I s u z s u =( ' # s u " ( ' )) . This result is a direct consequence of the variational principle (1). Moreover, the prop erties 1. and 2. determine completely the functions s u " . To measure the separation b etween invariant curves, wewilluse the Melnikov potential , i.e., a primitive of the Melnikov function, that is nothing else that the action P L ( ' n ' n +1 " )avaluated on the unp erturb ed separatrices ; := ; + ; ; up to the rst order in " . Prop osition 2.2 Let L :; ! R be the function denedby L ( z )= X n 2 Z L 1 ( ' n ' n +1 ) z n =( ' n v n )= F n 0 ( z ) : (10) Then: (i) L is wel l-de ned (the sum in (10) is absolutely convergent), analytic and F 0 -invariant (i.e., L F 0 = L ). (ii) If L is not identical ly constant and " is smal l enough, the invariant curves W u " , W s " intersect along a nite contact homo clinic orbit of F " , and F " is non-integrable.
92 A. Delshams and R. Ramrez-Ros Pro of of the Prop osition. Weonlysketch here the pro of. For more details, see 4]. (i) The sum is absolutely convergent since any orbit in the separatrix ; tends to z 1 at an exp onential rate as j n j!1 and L 1 (0 0) = 0. Thus L is well-dened and analytic. Moreover, a shift in the index of the sum do es not changes its value, so the F 0 -invariance of L is proved. (ii) Our measure of the distance b etween the p erturb ed asymptotic manifolds is the dierence of b er co ordinates (the v 's) $ ( ' " ):= d u " ( ' ) d ' ; d s " ( ' ) d ' = "M ( ' )+ O ( " 2 ) : M gives in rst order in " the distance b etween invariant curves, and thus can b e considered as a Melnikov function. For a primitive of $ with resp ect to ' ,$ := d = d ' ,we have the following expression: ( ' " ):= u " ( ' ) ; s " ( ' )= X n 2 Z L " ' ( n ) n ' ( n ) n +1 where ( ' s u n v s u n )= F n " ( ' d s u " ( ' ) = d ' ), ( n )=s,if n 0and ( n )=u, if n< 0. Let L :0 ] ! R the analytic function dened by L ( ' ):= L ( ' d 0 ( ' ) = d ' ), where 0 ( ' )= ; cos ' . Using the uniform approximations of W s u " by;andthevariational principle, it turns out that ( ' " )= "L ( ' )+ O ( " 2 ) : Thus, M =d L = d ' ,and now it is clear that if L is not identically constant, the same will happ en to L . Finally, one can see that M is not zero, analytic, and has a zero of nite order, using the fact that L ' is h -p erio dic, where ( t )=( ' ( t ) v ( t )). Consequently, for " small enough, the invariant curves W u " , W s " intersect along a nite contact homoclinic orbit of F " . By a result of Cushman 5], F " is non-integrable. 2 From nowon,we consider only ; + . Using the parameterization provided by (2.1), we can write ; + = f + ( t )=( ' ( t ) v ( t )) g ,and consider the Melnikovpotential L ( t )= X n 2 Z f ( t + hn ) with f ( t )= L 1 ( ' ( t ) ' ( t + h ))
On Birkho 's conjectureabout convex bil liards 93 where for simplicity of notation, wewrite L instead of L ,with L 1 as given in equation (8). Using formula (4), we arriveat an equivalent expression for f : f ( t ) = sech( t + h 2 )sech( t ) 1 ( ' ( t )) + sech( t + h ) 1 ( ' ( t + h ))] : (11) 2.2. Non-integrable billiards The aim of this pap er is to prove the following result. Theorem 2.1 Let f C " g be any non-trivial symmetric entireperturbation of an el lipse. Then the bil liardin C " is non-integrable for 0 < j " j 1 . Pro of of the Theorem. By prop osition 2.2, we only haveto prove that L is non constant. To thisend,we will use the following result. Lemma 2.2 Inacomplex neighbourhood D of t = i = 2 ,thereexists a analytic function G such that L ( t ) reads as L ( t )= 2 a b sech( t + h= 2) sech( t ; h= 2) 1 ( ' ( t )) + G ( t ) : Accepting for the moment this lemma, and using the fact that sech( t + h= 2) sech( t ; h= 2) is analytic non zero on t = i = 2, it turns out that t = i = 2 is a singular p ointof L ( t ) if and only if the same happ ens to 1 ( ' ( t )). But byhyp othesis, 1 is a - p erio dic entire function, and by lemma 2.1, sin ' ( t ) = sech( t )and cos ' ( t ) = tanh ( t )have simple p oles at t = i = 2 and no more singularities on = t = = 2. Since 1 is non constant, t = i = 2isa singular p ointof 1 ( ' ( t )) and consequently L ( t ) is not a constant. 2 Pro of of Lemma2.2. Lo oking at the expression (11) of f ,and using the fact that 1 is entire, the only p ossible singularities of f with = t = = 2 are t = i = 2, t = i = 2 ; h ,and t = i = 2 ; h= 2. Nevertheless, t = i = 2 ; h= 2isaremovable singularity, since it is a simple p ole of sech( t + h= 2), and also a zero of sech( t ) 1 ( ' ( t )) + sech( t + h ) 1 ( ' ( t + h )). So we can write L ( t )= X n 2 Z f ( t + nh )= f ( t ; h )+ f ( t )+ G 1 ( t )