JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 41, NUMBER 8
AUGUST 2000
Lagrangian submanifolds in product symplectic spaces S. Janeczkoa) Institute of Mathematics, Warsaw University of Technology, Pl. Politechniki 1, 00-661, Warsaw, Poland
共Received 26 January 1999; accepted for publication 13 April 2000兲 We analyze the global structure of Lagrangian Grassmannian in the product symplectic space and investigate the local properties of generic symplectic relations. The cohomological symplectic invariant of discrete dynamical systems is generalized to the class of generalized canonical mappings. Lower bounds for the number of two-point and three-point symplectic invariants for billiard-type dynamical systems are found and several examples of symplectic correspondences encountered from physics are presented. © 2000 American Institute of Physics. 关S0022-2488共00兲01608-X兴
I. INTRODUCTION
Let (M , ) be a symplectic manifold. We consider the product M ⫻M endowed with the symplectic structure ⍀⫽ 2* ⫺ * 1 , where i are the corresponding projections onto the components of M ⫻M . The space of Lagrangian submanifolds of (M ⫻M ,⍀) is a natural generalization of the group of symplectic transformations of (M , ). We notice that if :(M , ) →(M , ) is a symplectomorphism, then its graph, graph 傺M ⫻M is the Lagrangian submanifold, ⍀ 兩 graph ⫽0. There is an obvious motivation to study the global and local structure of such Lagrangian submanifolds, which are also called symplectic relations or symplectic correspondences 共cf. Ref. 1兲. They are coming from various branches of mathematics in which the symplectic ideas and methods were succesfully applied 共cf. Refs. 1–5兲. The very elementary examples of symplectic relations, which are not the graphs of symplectomorphisms, play an important role in geometrical diffraction theory 共Ref. 6兲. Consider the set M of all oriented affine lines in R3 . M is a four-dimensional symplectic manifold, M ⬅T * S 2 , constructed by the symplectic reduction from the free particle Hamiltonian hypersurface 共cf. Ref. 7兲. This is the space of rays of geometrical optics. One of the most classical systems that provide the nontrivial symplectic relation is a billiard system. If V is a smooth compact, convex region in Rn and X denotes its boundary hypersurface, then the symplectic relation joining the incoming 共going through the interior of the region兲 and outgoing rays, by the reflection law y⫽x⫺2(x 兩 n)n 共where n is the unit outer normal to X, and (• 兩 •) denotes the scalar product in Rn 兲, is a graph of symplectomorphism called the billiard map 共cf. Ref. 8兲. However, if V is no longer convex, then the reflection law should be extended by the diffraction role 共cf. Ref. 6兲, which prescribe to one incoming ray the family of outgoing rays gliding in the tangential point of an incoming ray. The billiard symplectic relation is no more the graph of a symplectomorphism. In the case of incoming ray, say x, going through the one-dimensional edge of an aperture in R3 , the outgoing rays y form a cone defined by the equations (x⫺y 兩 ␥ )⫽0, 兩 x 兩 ⫽ 兩 y 兩 , where ␥ is a vector tangent to an edge oriented according to the incoming ray orientation. Our aim in this paper is to provide the geometric framework for the action of generalized mechanical 共like nonconvex billiard兲 and optical 共like diffraction on apertures兲 systems. An important object in investigation of geometry of Lagrangian submanifolds is the manifold ⌳ n of linear Lagrangian spaces in 2n-dimensional symplectic space, called the Lagrangian Grassmannian 共cf. Ref. 9兲. Its natural stratification, allowing us to indicate the global structure of a兲
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© 2000 American Institute of Physics
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J. Math. Phys., Vol. 41, No. 8, August 2000
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Lagrangian submanifolds and their singularities, is constructed as follows. Let us fix ␣ 苸⌳ n . By ⌳ ␣n , we denote the set of all Lagrangian subspaces in ⌳ n that are not n ␣ ⌳ n,k , transversal to ␣. We have ⌳ n␣ ⫽艛 k⫽1 ␣ ⫽ 兵  苸⌳ n ;dim共  艚 ␣ 兲 ⫽k 其 ⌳ n,k
and ␣ ⫽ dim ⌳ n,k
n 共 n⫹1 兲 k 共 k⫹1 兲 ⫺ ⭓3, 2 2
共1兲
if k⬎1. ⌳ ␣n may be oriented by choosing the vectors from T ⌳ ␣ ⌳ n transversal to ⌳ ␣n , formed by n symmetric bilinear quadratic forms on elements 苸⌳ n␣ , which are positive definite on ␣ 艚. So ⌳ n␣ with this orientation represents a singular cycle that is Poincare´ dual to the universal Maslov class 共cf. Ref. 10兲. Now we pose the following problem: Does there exists a similar (to that of the standard classification of Lagrangian singularities in a cotangent bundle) classification of Lagrangian submanifolds and their singularities in the product symplectic space exploiting the canonical product structure? Approaching the answer for this question we investigate the canonical stratification 共i.e., a partition into smooth submanifolds as it was done for ⌳ n above兲 of the Lagrangian Grassmannian ⌳ 2n in the product symplectic space, induced by the product structure. This stratification naturally appears in the theory of linear symplectic relations and is especially important for searching the geometric structure of the images by symplectic relations 共cf. Refs. 7 and 11兲. In Sec. II we prove that any linear symplectic relation in the product space is a composition of reduction relation and a symplectomorphism. By this decomposition property ⌳ 2n is stratified and the codimension formulas are calculated. In Sec. III, the first step into the theory of classification of germs of nonlinear symplectic relations is done, and the generic appearance of some singular points is proved. In the last section the action of ⌳ 2n onto elements of ⌳ n is considered in the framework of unitary group U(2n) and homogeneous space U(2n)/O(2n) representing ⌳ 2n 共cf. Refs. 10 and 9兲. In the nonlinear case of symplectic relations the iterational cohomological symplectic invariant was introduced and its cohomological properties were described. By the Morse theory approach 共cf. Refs. 12 and 13兲, the number of two-point and three-point 共defined on two-point and three-point periodic orbit of symplectic relation兲 symplectic invariants were estimated from below for a possibly nonconvex billiard system and systems of equally charged particles on the surface. To conclude, we note that this paper had its origin in an attempt to find the possible complete classification of symplectic invariants by action of generalized symplectic mappings 共cf. Ref. 7兲. The results here show that there is an open area for such invariants with applications to classical physical problems.
II. LAGRANGIAN GRASSMANNIAN IN THE PRODUCT SYMPLECTIC SPACE
Now we consider the linear product symplectic space, M⫽ 共 M ⫻M , 2* ⫺ * 1 兲, where (M , ) is a 2n-dimensional symplectic vector space. By ⌳ 2n we denote the Lagrangian Grassmannian of linear subspaces in M. By M 1 and M 2 we denote the symplectic spaces canonically placed in M, M 1 ⫽M ⫻ 兵 0 其 , M 2 ⫽ 兵 0 其 ⫻M . Equivalently, we write 共 M 1 ⫻M 2 , 2* 2 ⫺ 1* 1 兲 ,
for M, where
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J. Math. Phys., Vol. 41, No. 8, August 2000
2⫽ * 2 ⫺* 1 兩 兵 0 其 ⫻M ,
S. Janeczko
⫺ 1 ⫽ 2* ⫺ 1* 兩 M ⫻ 兵 0 其 .
At first, we have the natural decomposition. Lemma II.1: If L苸⌳ 2n , then L is transversal to M 1 and to M 2 simultaneously, or L is not transversal to M 1 and L is not transversal to M 2 . Proof: If L is transversal to M 2 then it may be parametrized by M 1 so L is a graph of a maximal rank symplectic mapping M 1 →M 2 and so has to be transversal to M 1 共one can replace M 2 by M 1 in this argument兲. If L is not transversal to M 1 , then assuming that L is transversal to M 2 on the basis of the previous argument we get the transversality of L to M 1 , which contradicts to our assumption. 䊐 By the critical subset of ⌳ 2n we denote the set C⌳ 2n of those Lagrangian subspaces of M, which are not transversal simultaneously to both subspaces M 1 and M 2 : C⌳ 2n ⫽ 兵 L苸⌳ 2n :L is not transversal to M 1 and L is not transversal to M 2 其 . Elements of C⌳ 2n cannot be obtained as the graphs of linear symplectic transformations between M 1 and M 2 . By supercritical set of ⌳ 2n we denote the Cartesian product, S⌳ 2n ⫽⌳ n ⫻⌳ n . The elements of this set are Lagrangian subspaces L⫽(W 1 ,W 2 ), where W 1 and W 2 are Lagrangian subspaces in (M 1 , 1 ) and (M 2 , 2 ), respectively. By the formula dim ⌳ n ⫽n(n⫹1)/2, we find codim S⌳ 2n ⫽n 2 . If R 1 傺(M 1 ⫻M 2 , 2* 2 ⫺ 1* 1 ) is a Lagrangian subspace 共linear symplectic relation兲, then we define the corresponding transpose Lagrangian subspace R t1 in (M 2 ⫻M 1 , 1* 1 ⫺ * 2 2 ), R t1 ⫽ 兵 共 v 2 , v 1 兲 苸M 2 ⫻M 1 ; 共 v 1 , v 2 兲 苸R 其 . If we have another Lagrangian subspace, say R 2 in the product space (M 2 ⫻M 3 , 3* 3 ⫺* 2 2 ), then we define the composition of R 1 and R 2 , R 2 ⴰR 1 as the following Lagrangian subspace: R 2 ⴰR 1 ⫽ 兵 共 v 1 , v 3 兲 苸M 1 ⫻M 3 ;᭚ v 2 苸M 2 共 v 1 , v 2 兲 苸R 1 , 共 v 2 , v 3 兲 苸R 2 其 , in the product symplectic space (M 1 ⫻M 3 , * 3 3⫺ * 1 1 ). Proposition II.1: If L苸C⌳ 2n then L has the following decomposition: ˜ ⴰR 1 , L⫽R t2 ⴰL where ˜L , R 1 , R 2 are linear Lagrangian subspaces: ˜L 傺 共 M ˜ 1 ⫻M ˜ 2 ,* ˜ 2 ⫺ 1* ˜ 1兲, 2
˜ 1 ,* R 1 傺 共 M 1 ⫻M ˜ 1 ⫺ 1* 兲 , 2
˜ 2 , 2* ˜ 2⫺ * R 2 傺 共 M 2 ⫻M 1 兲, ˜ 1 and M ˜ 2 , respectively, and R 1 , R 2 are graphs of projections 1 and 2 onto M
1* ˜ 1 ⫽ 兩 1 (L) ,
2* ˜ 2 ⫽ 兩 2 (L) .
˜ 1, ˜ 2 are defined uniquely by the above formulas, and ˜L 苸⌳ 2n⫺2k The symplectic forms ⫺C⌳ 2n⫺2k , for some k苸N.
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J. Math. Phys., Vol. 41, No. 8, August 2000
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Proof: If L苸C⌳ 2n then, by Lemma II.1 we have 1 (L)債V 1 , 2 (L)債V 2 , where V 1 , V 2 are hypersurfaces in M 1 and M 2 , respectively. If there is an equality above then V 1 and V 2 are ⬔ coisotropic, so we have the natural projections i along the symplectic polars V ⬔ 1 傺V 1 , V 2 傺V 2 ˜ 1 ⫽(V 1 /V ⬔ ˜ 2 ⫽(V 2 /V ⬔ onto the symplectic reduced spaces M ˜ 1 ), M ˜ 2 ). The symplectic polar 1 , 2 , to the subspace V傺(M , ) is defined to be the subspace V ⬔ ⫽ 兵 v 苸M ; ( v ,u)⫽0,᭙ u苸V 其 . So we ˜1 represent L uniquely by two hyperspaces V i and the Lagrangian subspace ˜L 苸⌳ 2n⫺2 in (M ˜ ˜ ⫻M 2 , 2* ˜ 2 ⫺ 1* ˜ 1 ). If L 苸C⌳ 2n⫺2 , then we may proceed in an analogous way and obtain the 䊐 noncritical representation for ˜L . Example II.1: If n⫽2 we have only two strata of the singular set C⌳ 4 : Elements of the first maximal stratum C 1 ⌳ 4 are determined by the pairs of two coisotropic subspaces, V 1 in M 1 and V 2 in M 2 and the symplectic linear maps between the corresponding reduced symplectic spaces. It is easy to calculate dim C 1 ⌳ 4 ⫽9. The second stratum is the supercritical set S⌳ 4 , and its dimension dim S⌳ 4 ⫽6. In general, we have the following result concerning the structure of the singular set C⌳ 2n . Theorem II.1: There is the following partition of the singular set C⌳ 2n into the smooth submanifolds, n
C⌳ 2n ⫽ 艛 C k ⌳ 2n , k⫽1
where the elements of C k ⌳ 2n are determined by the pairs of two coisotropic subspaces V 1 in M 1 and V 2 in M 2 of codimension k and the symplectic linear automorphism of the 2n⫺2k-dimensional symplectic space. In this partition C n ⌳ 2n ⫽S⌳ 2n . Proof: In fact, it follows from the property that the projection of L苸⌳ 2n onto M 1 and M 2 is always coisotropic 共or Lagrangian兲. Thus, starting from the hypersurfaces we see that the corresponding ˜L 苸⌳ 2n⫺2 , in the product of reduced symplectic spaces 共as it was proved in the Proposition II.1兲, projects onto these spaces or onto their hypersurfaces in the more degenerated case. Repeating this argument for further representations of L, we get the natural decomposition by equally dimensional coisotropic subspaces and linear symplectic maps in, respectively, smaller dimensional symplectic space. 䊐 Corollary II.1: codim C k ⌳ 2n ⫽k 2 ,
k⫽1,...,n.
Proof: We calculate the dimension of the isotropic Grassmannian I 2n k of k-isotropic planes in 2n-dimensional symplectic space V 共cf. Ref. 14兲, 1 dim I 2n k ⫽2nk⫺ 2 k 共 3k⫺1 兲 .
This is the dimension of the corresponding space of 2n⫺k-dimensional coisotropic subspaces in V. Since dim ⌳ 2n ⫽2n 2 ⫹n, we get codim C k ⌳ 2n ⫽dim ⌳ 2n ⫺2 dim I 2n k ⫺dim共 ⌳ 2n⫺2k 兲 ⫽n 共 2n⫹1 兲 ⫺2„2nk⫺ 21 k 共 3k⫺1 兲 …⫺ 共 n⫺k 兲共 2n⫺2k⫹1 兲 ⫽k 2 . In comparison to the inequality 共1兲, we have
䊐
codim C k ⌳ 2n ⭓4, if k⬎1.
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J. Math. Phys., Vol. 41, No. 8, August 2000
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III. LOCAL CLASSIFICATION OF SYMPLECTIC RELATIONS
Let (L, p) be germ of a symplectic relation 共Lagrangian submanifold兲 in M. Now we introduce the natural equivalence group acting on the space of such germs. Definition III.1: We say that the two germs (L 1 ,p 1 ), (L 2 ,p 2 ) of symplectic relations in M are equivalent if there exist two symplectomorphism germs B 1 :„M 1 , 1 (p 1 )…→„M 1 , 1 (p 2 )… and B 2 :„M 2 , 2 (p 1 )…→„M 2 , 2 (p 2 )… such that the symplectomorphism B 1 ⫻B 2 of M sends L 1 into L 2 and p 1 into p 2 . For the symplectic relation L傺M, we define the corresponding symplectic Gauss map, G:L苹 p→T p L苸⌳ 2n . We call L to be, in general, position 关or generic 共cf. Ref. 15兲兴 if G is transversal to C⌳ 2n n ⫽艛 k⫽1 C k ⌳ 2n . We say that L has a k-vertical position at p苸L if G(p)苸C k ⌳ 2n . We call k a rank of k-vertical position. A 0-vertical position corresponds to the case when L is a graph of a local symplectomorphism at p, i.e., G(p)苸⌳ 2n ⫺C⌳ 2n . For generic L the isolated points of vertical position appear only if n⫽2s 2 , for some s苸N. In this case they are isolated points in the 2s-vertical position. In their neighborhood there are k-vertical positioned points with k⭐2s. Following the standard representation of Lagrangian germs 共cf. Ref. 16兲 we have the following preparatory lemma. Lemma III.1: For any germ of a Lagrangian submanifold (L,p)傺M there are local cotangent bundle structures around 1 (p), say T * X and around 2 (p), say T * Y , such that (L,p) is generated in M⬵ 共 T * X⫻T * Y , 2* Y ⫺ 1* X 兲 , by a germ of a generating function F:(X⫻Y , X⫻Y (p))→R, such that, in local coordinates on (X⫻Y , X⫻Y (p)) we have n
F 共 x,y 兲 ⫽
兺
i j⫽1
x i y j i j 共 x,y 兲 ,
共2兲
where X and Y are the corresponding Liouville symplectic structures on T * X and T * Y , respectively. ˜ )… are Darboux coordinates on M, then we find the partition I艛J Proof: If „(p,q),( ˜p ,q ˜ ⫽ 兵 1,...,n 其 , ˜I 艚J ˜ ⫽0” , such that there exists a smooth function ⫽ 兵 1,...,n 其 , I艚J⫽0” , ˜I 艛J S(p I ,q J , ˜p˜I ,q ˜ ˜J ), which is a generating function for (L, p) 共cf. Refs. 5 and 16, Sec. III.19.3兲. By the symplectomorphism ˜ ,q ˜ 兲 ⫽ 共 ⫺q I ,p J ,p I ,q J ;⫺q ˜ ˜I ,p ˜ ˜J ,p ˜ ˜I ,q ˜ ˜J 兲 ⫽ 共 ,x; ,y 兲 , ⌽ 共 p,q;p which preserves the product structure of M, we find the generating function F(x,y) for (L,p) in the canonical special symplectic structure T * X⫻T * Y on M. The coordinates ( ,x)苸T * X, ( ,y)苸T * Y are new coordinates on the cotangent bundles in which (L, p) is generated by the generating function F. Then, further on, using the symplectomorphisms of M 1 and M 2 preserving the corresponding cotangent bundle structures, we obtain the reduced form 共2兲 of function F. We recall that L is described by the following equations:
i⫽
F 共 x,y 兲 , yi
j ⫽⫺
F 共 x,y 兲 , x j
1⭐i, j⭐n.
䊐 Theorem III.1: Let p苸L and we assume that the Lagrangian submanifold L, around p, is generated by the generating function in the normal form (2). Then we have the following.
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J. Math. Phys., Vol. 41, No. 8, August 2000
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共1兲 Rank of the vertical position of L at p is equal to the corank of the matrix ( i j ⫽ 2 F/ x i y j ) at X⫻Y (p). 共2兲 At each p苸L, for a generic L, the family of mappings,
冉兺 n
⌽:X⫻Y→Rn ,
⌽ 共 x,y 兲 ⫽
n
j⫽1
y j 1 j 共 x,y 兲 ,...,
兺
j⫽1
冊
y j n j 共 x,y 兲 ,
where i j (x,y) are defined in Eq. (2), has a generic singularity at X⫻Y (p). Proof: 共1兲 Any linear relation L, by Lemma III.1, is equivalent to one generated by the quadratic form 兺 inj⫽1 x i y j a i j . So the dimension of the kernel of the matrix (a i j ) is exactly equal to the rank of verticality of L. This is a local symplectic invariant of (L,p) that does not depend on the choice of the corresponding cotangent bundle structures. 共2兲 By Lemma III.1, any relation L is locally generated by the generating function F(x,y) ⫽ 兺 i j x i y j i j (x,y), and by the form of function F uniquely represented by a smooth family of mappings, ˆ 1 共 x,y 兲 ,..., ˆ n 共 x,y 兲 …, ⌽ 共 x,y 兲 ⫽„ such that ⌽(x,0)⬅0. We see that the Gauss map G:L→T p L corresponds exactly to the one-jet extension j 1 ⌽(x,y) of the mapping ⌽, so the transversality of G is equivalent to the corresponding transversality of ⌽ to the canonical stratification of smooth mappings of Rn ⫻Rn into Rn 共cf. Refs. 15, 16兲. 䊐 Corollary III.1: At any point p苸L of the 0-vertical position of L, symplectic relation L is parallelizable, i.e., it is locally symplectically equivalent to its tangent space T p L with the following generating function: n
F 共 x,y 兲 ⫽
兺 xiy i .
i⫽1
Remark III.1: If n⫽2 then the supercritical points appear in generic L as the isolated points, in fact codim C 2 ⌳ 4 ⫽4, and G is transversal to S⌳ 4 ⫽C 2 ⌳ 4 (see Example II.1). If p苸L is a supercritical transversal point then on the basis of Lemma III.1, on a neighborhood of p, L is generated locally by the following generating function: 2
F 共 x,y 兲 ⫽
兺
i j⫽1
x i y j i j 共 x,y 兲 ,
where i j (0,0)⫽0, i j⫽1,2, p⫽0, and the transversality condition is equivalent to rank D⌽ 共 0 兲 ⫽4, where ⌽(x,y)⫽„ i j (x,y)…苸M 2⫻2 . If we need to iterate a symplectic relation L we have to use the symplectic equivalence group preserving the canonical product structure of M⫽(M ⫻M , 2* ⫺ 1* ). We say that the two ˜ i , i⫽1,2, are D-equivalent 共diagonal germs (L 1 , p 1 ), (L 2 , p 2 )傺M, where 1 (p i )⫽ 2 (p i )⫽p equivalence兲 if there exists a symplectomorphism germ B:(M ,p ˜ 1 )→(M ,p ˜ 2 ) such that (B⫻B) ⫻(L 1 )⫽L 2 . Using the notation of the composition of symplectic relations, we can write L 2 ⫽Bˆ ⴰL 1 ⴰBˆ t . Now slightly extending the proof of Lemma III.1, we have the following result.
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Lemma III.2: For any germ (L, p)傺M there exists a local cotangent bundle structure T * X (D equivalence) around 1 (p), such that (L,p) is generated in M⬵ 共 T * X⫻T * X, 2* X ⫺ 1* X 兲 , by a Morse Family germ F:(X⫻X⫻R k ,0)→R 关we assumed 1 (p)⫽0兴, n
F 共 x,y, 兲 ⫽
兺 x i i共 x,y, 兲 ,
共3兲
i⫽1
such that k⭐dim X. If the integer k is minimal then it is an invariant of D-equivalence symplectic group action. We see that the linear symplectic relations in M are classified by the classes of linear mappings, ⌽⫽ 共 1 ,..., n 兲 :Rn ⫻Rn ⫻Rk →Rn , extracted from the normal form 共3兲, with the standard equivalence relation ⌶: 共 x,y, 兲 →„A 共 x 兲 ,Y 共 y 兲 ,⌳ 共 x,y, 兲 …, where the equivalent mapping, say ⌽ ⬘ , is given by ⌽ ⬘ 共 x,y, 兲 ⫽A T ⌽„A 共 x 兲 ,B 共 y 兲 ,⌳ 共 x,y, 兲 …. Remark III.2: An important source of the nontrivial symplectic relations is given by the KdV and MKdV hierarchies of nonlinear differential equations (cf. Ref. 2). The main example is the Darboux relation (correspondence). If we consider the Schro¨dinger operator, L⫽⫺D 2 ⫹u⫽⫺ 共 D⫹ f 兲共 D⫺ f 兲 ⫽⫺D 2 ⫹ 共 f 2 ⫺ f ⬘ 兲 , and its isospectral deformation, ˜L ⫽⫺D 2 ⫹ v ⫽⫺ 共 D⫺ f 兲共 D⫹ f 兲 ⫽⫺ 共 D⫹g 兲共 D⫺g 兲 ⫽⫺D 2 ⫹g ⬘ ⫹g 2
冉
D⫽
冊
d , dx
then we get the Darboux relation ⫺ f ⬘ ⫹ f 2 ⫽g ⬘ ⫹g 2 , in the product of two copies of function space F of variable x, endowed with the difference of the Poisson structures,
兵 F,G 其 ⫽
冕 ␦␦
F ␦G D dx, f ␦f
where F 关 f 兴 ,G 关 f 兴 are the functionals on F. For the Schro¨dinger operator ⫺D 2 ⫹u there is an infinite hierarchy of isospectral flows, with a corresponding set of integrals I k 关 u 兴 ⫽ 兰 L k (u,u x ,u xx ,...)dx. The first integrals expressed by f -variable can be written in the form H k关 f 兴 ⫽ H 0关 f 兴 ⫽
冕
冕
共 f x ⫹ f 2 兲 dx,
Lk 共 f , f x , f xx ,... 兲 dx; H 1关 f 兴 ⫽
1 2
冕
共 f 2x ⫹2 f x f 2 ⫹ f 4 兲 dx,
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J. Math. Phys., Vol. 41, No. 8, August 2000
H 2关 f 兴 ⫽
冕冉
Lagrangian submanifolds in the product . . .
冉
1 2 3 f ⫹5 f 2 f 2x ⫹ f 6 ⫹ f f 2x ⫹ f 5 2 xx 5
冊冊
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dx.
x
The Darboux transformation preserves these integrals and gives the difference of the integrands that is the x derivative, i.e., Lk 关 f 兴 ⫺Lk 关 g 兴 ⫽
d ¯F 关 f ,g 兴 , dx k
共4兲
where ¯F k 关 f ,g 兴 are differential polynomials. Now we restrict the problem to n jets of functions of (n) ) and provide the symplectic reduction to finite-dimensional symplectic x, so L⫽L( f , f x ,..., f x...x space by reduction of functional parameter; ␦ H/ ␦ f ⫽0, which gives the finite-dimensional reduced space and indicates the symplectic structure from the formula dL⫽
␦H d p i dq i , df⫹ ␦f dx 兺 i
where q 1 ⫽ f , q 2 ⫽ f x ,...,q n ⫽ f (n⫺1) . Thus, after restriction of Darboux transformation f →g to the ‘‘stationary hypersurface’’ ␦ H/ ␦ f ⫽0, on the basis of (4), we get the symplectic relation d dx
冉兺 i
p i dq i ⫺
兺j P j dQ j
冊
⫽
d ¯, dF dx
generated by the Morse family S:
兺i p i dq i ⫺ 兺j P j dQ j ⫽dS. As an example one can consider the integrand (cf. Ref. 2), L⫽ 12 共 f 2x ⫹ f 4 兲 ⫹ 21 a f 2 ⫹b f , where a, b are real parameters. In the symplectic variables q⫽ f , p⫽ f x , Q⫽g, P⫽g x , the corresponding Darboux relation is generated by the function S 共 q,Q 兲 ⫽ 31 共 q 3 ⫹Q 3 兲 ⫹ 21 a 共 Q⫹q 兲 ⫹b log共 q⫹Q 兲 . IV. ACTION AND ITERATION OF SYMPLECTIC RELATIONS
Let L傺M be a symplectic relation and S傺(M , ) be a subset of M . Then we define the image of S by L; L 共 S 兲 ⫽ 兵 p苸M :᭚ p ⬘ 苸S , 共 p ⬘ ,p 兲 苸L 其 . Obviously this action of L on subsets of M preserves all their symplectic properties. Thus, if S is Lagrangian, isotropic, or coisotropic, then L(S) is also Lagrangian, isotropic, or coisotropic, respectively, unless it is singular 共cf. Refs. 7, 17, and 18兲. Now we consider the linear case, and assume S is a Lagrangian subspace. As a canonical pair we define (l,L), where L苸⌳ 2n and l苸⌳ n . Proposition IV.1: There is a natural mapping, H:⌳ n ⫻⌳ 2n →⌳ n ,
H 共 l,L 兲 ⫽L 共 l 兲 .
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Proof: We have to check that L(l) is a Lagrangian subspace of (M , ). Indeed, we can choose the cotangent bundle fibration on M and represent l and L by generating families, say (,q 1 )→G(,q 1 ) for l and ( ,q 1 ,q)→F( ,q 1 ,q) for L. Then the generating family for the image L(l) is defined by H 共 , ,,q 兲 ⫽F 共 , ,q 兲 ⫹G 共 , 兲 . By the standard reduction of 共,,兲-Morse parameters, we get the generating family for L(l) in the form 共cf. Refs. 16 and 5兲 k
˜ 共 ,q 兲 ⫽ f 共 q 兲 ⫹ H⬵F
兺 ig i共 q 兲 ,
i⫽1
where f is a quadratic form and g(q)⫽„g 1 (q),...,g k (q)… is a linear mapping, g(q)⫽Aq. Now we easily see that the space
再
共 p,q 兲 苸M :᭚ 苸Rk :p⫽
˜F ˜F 共 ,q 兲 ,0⫽ 共 ,q 兲 q
冎
is an n-dimensional Lagrangian subspace of (M , ) because dim(Ker A)⫹dim(Im A)⫽n. 䊐 Another view on the image L(l) is given through the unitary group reconstruction of Lagrangian subspaces 共cf. Ref. 9兲. We write
L⫽
再 冉 冊冉 A B
冊
␣1 • , ␣ 2n
冎
␣ i 苸R ,
where
冉冊
A 苸U 共 2n 兲 . B
The corresponding projections onto the first and the second component of M are given in the form
再冉 冊 冎 再冉 冊 冎
␣1 L 共 M 兲 ⫽ A • , ␣ i 苸R ⫽ 1 共 L 兲 , ␣ 2n t
L共 M 兲⫽ B If l is given in the form
␣1 • , ␣ i 苸R ⫽ 2 共 L 兲 . ␣ 2n
再冉 冊 冎
1 l⫽ G • ,  i 苸R , n where G苸U(n). Then, at first, we define the subspace V (A,G) ,
再
冉 冊 冉 冊冎
␣1 1 R 傻V (A,G) ⫽ ␣ :᭚  苸Rn A • ⫽G • ␣ 2n n 2n
,
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J. Math. Phys., Vol. 41, No. 8, August 2000
Lagrangian submanifolds in the product . . .
and finally L(l) is defined in the following way:
再冉 冊
5651
冎
␣1 : ; ␣ 苸V (A,G) , ␣ 2n
L共 l 兲⫽ B
where we denote ␣ ⫽( ␣ 1 ,..., ␣ 2n ), and  ⫽(  1 ,...,  n ). The mapping
:⌳ 2n 苹L→H 共 l 0 ,L 兲 ⫽L 共 l 0 兲 苸⌳ n , where l 0 is a fixed element of ⌳ n , is a fiber bundle projection. Let L be represented by ( BA ) 苸U(2n), where we denote by A⫽A 1 ⫹iA 2 , B⫽B 1 ⫹iB 2 the decomposition of A and B complex matrices. Proposition IV.2: The fiber of is defined by ( BA )苸U(2n), such that we have the following. A
共1兲 rank ( B 2 )⭐n. 2
B
共2兲 A 1 and B 1 are surjective on V⫽Ker( A 2 ). 2
Proof: It is enough to find such L苸⌳ 2n , that L(l 0 )⫽l 0 . Choose l 0 ⫽ 兵 (q⫹ip)苸M :p⫽0 其 , then we get the condition
冉 冊冉 A B
冊
␣1 : 苸R2n , ␣ 2n A
and surjectivity of A 1 and B 1 on the Kernel of ( B 2 ) corresponds to the property that l 0 2
⫽Rn 傺Cn is mapped by L onto l 0 . 䊐 Now using the Proposition II.1 we investigate the geometric structure of an image L(l). Proposition IV.3: Any image of l苸⌳ n by L苸⌳ 2n is a Lagrangian subspace l ⬘ of a coisotropic space L(M )⫽ 2 (L)傺(M , ). Thus, it is a counterimage, by a canonical reduction
L(M )→ L(M )/L(M ) ⬔ of some Lagrangian subspace in the reduced symplectic space ˜ …, * ˜ ⫽ 兩 L(M ) . „L(M )/L(M ) ⬔ , Proof: We define L t (M )⫽ 1 (L), L(M )⫽ 2 (L), which are the coisotropic subspaces of the same codimension in both components of M. L(M ) is a coisotropic space, so the Kernel of 1 兩 L projected onto L(M ), by 2 , is a symplectic polar of L(M ), and vice versa the Kernel of 2 兩 L projected onto L t (M ), by 1 , is a symplectic polar of L t (M ). Thus, for the pair (l,L), in any common position, the image L(l) is a Lagrangian subspace of the coisotropic space L(M ). 䊐 Remark IV.1: n C k ⌳ 2n , where GSp 2n denotes 共1兲 The mapping is smooth on all strata of ⌳ 2n ⫽GS p 2n 艛艛 k⫽1 the graphs of symplectomorphisms. Only on these strata can we pull back the universal
det 2
Maslov class 关 * 兴 , (⌳ 2n 傻GSp 2n → ⌳ n → S 1 ) (cf. Refs. 10 and 9). 共2兲 In the smooth nonlinear case, L(l) is always isotropic on his smooth strata. The corresponding local generating family is of the form (cf. Ref. 7) H共,,,q兲⫽G共,兲⫹F共,,q兲, where G(,q 1 ) is a generating Morse family for l and F( ,q 1 ,q) is a generating Morse family for L. We notice that H is not necessary Morse family so the corresponding image L(l) may not be smooth. There is an interesting symplectic invariant prescribed to the symplectic relation and based on the cohomological properties of the relation 共cf. Refs. 19 and 20兲. Now we will assume that the Lagrangian submanifold L傺M is compact 共with boundary兲, and the first cohomology group
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H 1 (L,R) is trivial. Instead of (M 1 , 1 ), (M 2 , 2 ) we take two copies of the same symplectic manifold (M , ). For any choice of 1-form ␣, such that ⫽d ␣ , the 1-form 2* ␣ ⫺ 1* ␣ 兩 L is exact and
* 2 ␣⫺* 1 ␣ 兩 L ⫽dH,
共5兲
for some smooth generating function H:L→R. If ␣ 1 is another choice of a 1-form for which d ␣ 1 ⫽ , then d( ␣ 1 ⫺ ␣ )⫽0 and ␣ 1 ⫺ ␣ ⫽dG, for some smooth function G:M →R 共where M has a boundary or is not compact兲. For a new underlying 1-form ␣ 1 , the Lagrangian submanifold L has another generating function H 1 :L→R, such that
2* ␣ 1 ⫺ 1* ␣ 1 兩 L ⫽dH 1 .
共6兲
Proposition IV.4: The generating functions H 1 and H defined by the formulas (6) and (5) are joined by the following relation: H 1 ⫽H⫹ 共 * 2 G⫺ 1* G 兲 兩 L .
共7兲
Proof: Subtracting formula 共5兲 from the formula 共6兲, by sides, we get
2* 共 ␣ 1 ⫺ ␣ 兲 ⫹ 2* 共 ␣ 1 ⫺ ␣ 兲 ⫽d 共 H 1 ⫺H 兲 , and because ␣ 1 ⫺ ␣ ⫽dG we get finally d 共 2* G⫺ * 2 G 兲 ⫽d 共 H 1 ⫺H 兲 . If we normalize the additive constants in the definitions of H, H 1 , and G, we obtain the formula 共7兲. 䊐 Now we consider the multiple, iterated images by the relation L. Let ⫽ 兵 (x 0 ,x 1 ) 苸L,(x 1 ,x 2 )苸L,...,(x k⫺1 ,x 0 )苸L, 其 . We will call the periodic orbit of L. We will associate with the following number: k⫺2
N共 兲⫽
兺
i⫽0
H 共 x i ,x i⫹1 兲 ⫹H 共 x k⫺1 ,x 0 兲 .
Now using the formula 共7兲 we have a natural property of N( ) 共cf. Ref. 19兲. Corollary IV.1: N( ) is an invariant with respect to the action of the group of symplectomorphisms operating on (M , ). If L⫽graph ⌽, where ⌽:(M , )→(M , ) is a symplectomorphism, then the set of numbers 兵 N( ) 其 , where is any periodic orbit of ⌽ is called the spectrum of ⌽. This spectrum is extensively studied if ⌽ is a billiard mapping associated with the convex region in Rn 共cf. Refs. 20 and 8兲. In the case of a graph, the formula 共7兲 reduces to the following one: H 1 ⫽H⫹⌽ * G⫺G, where ⌽ * ␣ ⫺ ␣ ⫽dH, and the iterational invariant to the periodic orbit of ⌽, ⫽ 兵 x 0 ,x 1 ,...,x k⫺1 其 is given in the form k⫺1
N共 兲⫽
兺
i⫽0
H共 xi兲.
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J. Math. Phys., Vol. 41, No. 8, August 2000
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As an example, we consider the billiard symplectic map. Let ⍀ be a smooth compact convex region in Rn . Let X be the boundary of ⍀ and T * X the cotangent bundle of X. The symplectic billiard map B:T * X→T * X is defined on the set U⫽ 兵 (x, )苸T * X: 兩 兩 ⬍1 其 . To the point (x, ) 苸U we prescribe the point (x ⬘ , ⬘ )苸U in the following way. Let n(x) be the outward unit normal vector to X at x. There is a unique, unit element ˜ 苸(Rn ) * such that 具 ˜ ,n(x) 典 ⬍0 and 具 ˜ , v 典 ⫽ 具 , v 典 for all v 苸T x X, where we identify T x X with the corresponding subspace in Rn . To a given (x, ) there exists x ⬘ 苸X, which is the unique point of intersection of X with the positive line segment; x ⬘ ⫽X艚 兵 x⫹t ,t⬎0 其 , where is a unique vector in Rn corresponding to ˜ , and ⬘ is defined as the unique element of T * x ⬘ X for which we have 具 , v 典 ⫽ 具 ⬘ , v 典 for all v 苸T x ⬘ X. Obviously B(x, )⫽(x ⬘ , ⬘ ) is symplectic and the generating function for graph B苸(T * X ⫻T * X, 2* X ⫺ 1* X ) is defined as the perimeter, ˜ 共 x,x ⬘ 兲 ⫽ 兩 x ⬘ ⫺x 兩 , H
˜ :X⫻X→R, H
where X is the Liouville one-form on T * X. By the projection X⫻X :T * X⫻T * X→X⫻X and the smooth map
:T * X→T * X⫻T * X,
共 x, 兲 ⫽ 共 x,⫺ ,x ⬘ , ⬘ 兲 ,
˜ such that B * X ⫺ X ⫽dH, which gives the * H we get the function H:T * X→R, H⫽ * X⫻X symplectic invariant k⫺1
N共 兲⫽
兺
i⫽0
H共 xi ,i兲,
for the periodic orbit ⫽ 兵 (x 0 , 0 ),...,(x k⫺1 , k⫺1 ) 其 , which is the length 兩 x 1 ⫺x 0 兩 ⫹¯⫹ 兩 x 0 ⫺x k⫺1 兩
of the closed geodesic of the billiard system and defines, for all closed geodesics, the length spectrum of ⍀ 共cf. Ref. 8兲. In general, if we assume that L傺M⫽ 共 T * X⫻T * X, 2* X ⫺ 2* X 兲 is generated by the smooth generating function F:(x,y)→F(x,y), then the invariant N( ) defined on the periodic orbit of L,
X⫻X 共 兲 ⫽ 兵 共 x 0 ,x 1 兲 , 共 x 1 ,x 2 兲 ,..., 共 x k⫺1 ,x 0 兲 其 is the critical value of the function G:X⫻...⫻X→R,
˜ 1 ,...,x ˜ k⫺1 兲 ⫽F 共˜x 0 ,x ˜ 1 兲 ⫹F 共˜x 1 ,x ˜ 2 兲 ⫹¯⫹F 共˜x k⫺1 ,x ˜ 0兲, G 共˜x 0 ,x
for which X⫻X ( ) is a critical point. Definition IV.1: By 兵 N( ) 其 k we denote the set of symplectic invariants for k-point periodic orbits. We will call them k-point symplectic invariants. For the general billiard system the Lagrangian submanifold L may not be the graph of any symplectomorphism. Let :X→Rn be an imbedding of a closed orientable surface and we assume that it is generic, i.e., the function ˜ 共 x,y 兲 ⫽ 兩 共 x 兲 ⫺ 共 y 兲 兩 , H defined on X⫻X outside of the diagonal ⌬ has only nondegenerate critical points on X⫻X⫺⌬. ˜ are, in fact, the two-point orbits or double normals of We easily see that the critical points of H
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the possibly nonconvex billiard system. The corresponding invariants are the critical values of ˜ . Using the Morse theory methods, Morse inequalities, 共cf. Refs. 13, 12, Theorem 1兲, we H obtain an estimation for the number of k-point invariants for small k. Theorem IV.1: If is a generic imbedding of a surface of genus g, then we have the following lower bound for the number of two-point symplectic invariants: # 兵 N 共 兲 其 2 ⭓2g 2 ⫹3g⫹3. For the generic billiards on the plane we have at least two two-point symplectic invariants. In the case of an ellipsoid, g⫽0, with the three unequal axes # 兵 N( ) 其 2 ⫽3. If this is an imbedding of the torus we have at least eight two-point symplectic invariants of the toruslike billiard system. In three-dimensional billiard systems, the lower bound for # 兵 N( ) 其 2 is expressed by the first Betti number d 1 of X. In fact, if :X→Rn is generic, then the lower bound is given by the numbers: 2d 21 ⫹3d 1 ⫹4,
if d 1 is even,
2d 21 ⫹3d 1 ⫹5,
if d 1 is odd.
or
If X⫽S n⫺1 then we have # 兵 N( ) 其 2 ⭓n. Now we consider an imbedding :X→Rn , which is generic with respect to the generating function V 共 x,y 兲 ⫽
1 , 兩 共 x 兲⫺ 共 y 兲兩
defined outside of the diagonal ⌬. The corresponding symplectic relation ˜L 傺M, defined by V , provides the geometric setting for finding the equilibrium positions of equally charged particles on an imbedded surface in Euclidean space. The iterational symplectic invariants N( ) define the least potential energy of the number of charged particles in equilibrium on X. The two-point symplectic invariants for ˜L exactly correspond to the double normals—equilibrium positions of the two equally charged particles—and thus the corresponding lower bounds are analogous to that established for the billiard system in Proposition IV.1. The three-point invariants are defined by the critical points of the function, V (3) 共 x,y,z 兲 ⫽
1 1 1 ⫹ ⫹ , 兩 共 x 兲⫺ 共 y 兲兩 兩 共 y 兲⫺ 共 z 兲兩 兩 共 z 兲⫺ 共 x 兲兩
defined outside the total diagonal ⌬ in X⫻X⫻X and generic. Now, following the further Morsetheory estimations obtained in Ref. 12 共Theorem 3兲, we get the following lower bound for the number of three-point symplectic invariants of ˜L . Theorem IV.2: If :X→Rn is a generic imbedding of a surface of genus g, then we have the following lower bounds: # 兵 N 共 兲 其 3 ⭓ 共 4g 3 ⫹8g 2 ⫹6g⫹12兲 /3,
for g⫽⫽2 共 mod 3 兲 ,
or # 兵 N 共 兲 其 3 ⭓ 共 4g 3 ⫹8g 2 ⫹6g⫹14兲 /3,
for g⫽2 共 mod 3 兲 ,
if g⫽⫽1 or # 兵 N( ) 其 3 ⭓11 if g⫽1.
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J. Math. Phys., Vol. 41, No. 8, August 2000
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ACKNOWLEDGMENTS
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