A Quaternionic Structure as a Landmark for Symplectic Maps
aa r X i v : . [ m a t h . S G ] O c t A Quaternionic Structure as a Landmark forSymplectic Maps
Hugo Jim´enez-P´erez ∗ October 31, 2019
Abstract
We use a quaternionic structure on the product of two symplectic man-ifolds for relating Liouvillian forms with linear symplectic maps obtainedby the symplectic Cayley’s transformation.
One of the main difficulties for constructing symplectic maps by the methodof generating functions is the resolution of the Hamilton-Jacobi equation. In-stead of solving such an equation, in this paper we consider a local quaternionicstructure on the symplectic product manifold and the three different symplecticforms induced by this structure. Symplectic maps are constructed using theprimitive Liouvillian forms related to these symplectic forms.
Let (
M, ω ) be a 2 n -dimensional symplectic manifold with symplectic form ω .A symplectomorphism is a diffeomorphism φ : ( M, ω ) → ( M, ω ) preservingthe symplectic structure φ ∗ ω = ω , where the star stands for the pull-back ofdifferential forms. When the symplectic structure has a global primitive linearform θ , then ( M, dθ ) is called an exact symplectic manifold . Main representativesare cotangent bundles ( T ∗ Q , dα ) which possesses a canonical or tautologicalform α called the Liouville form . We define a
Liouvillian form in an exactsymplectic manifold, as any representative θ ∈ [ α ] in Ω ( M ), and a Liouvillevector field Z by the implicit equation θ = ( i Z ω ).We are interested in symplectic maps for constructing symplectic integrators,then we can consider maps defined on convex balls B ⊂ ( M, ω ) containing atthe same time, the points z , z h and the full path of symplectic diffeomorphisms ∗ Funding: The author was supported by the
Fondation du Coll`ege de France , Total (RCNPU14150472), and the ERC Advanced Grant WAVETOMO (RCN 99285). The author de-clares that he has no conflict of interest. t = φ ( t ) connecting them z h = φ h ( z ). Then φ ([0 , h ]) ֒ → B is an embeddedsegment of curve. In a convex ball, there always exist primitive 1-forms θ byPoincar´e’s lemma and consequently we can apply this procedure locally on anysymplectic manifold. Define the product P = M × M of two copies of an exact symplectic manifold( M, ω = dθ ), which we denote by ( M , ω ) and ( M , ω ), respectively. Each copycorresponds to the flow of a (Hamiltonian) system at two different times t = 0and t = h for small h . The canonical projections π i : P → M i for i = 1 , θ ⊖ and ω ⊖ on P by, θ ⊖ = π ∗ θ − π ∗ θ , and ω ⊖ = π ∗ ω − π ∗ ω .It is well known that ( P , ω ⊖ ) is a symplectic manifold of dimension 4 n [6]. Thegraph of any symplectic map φ : ( M , ω ) → ( M , ω ), defined byΓ φ = { ( z, φ ( z )) ∈ P | z ∈ M , φ ( z ) ∈ M } , is a Lagrangian submanifold in ( P , ω ⊖ ).Consider Γ φ as an embedding : Λ ֒ → P with (Λ) = Γ φ ⊂ P , and by abuseof notation we identify Λ and its image (Λ). In terms of this embedding wehave ∗ ω ⊖ ≡
0. In addition to the symplectic form ω ⊖ , for every x ∈ P thereexists an induced endomorphism on T x P which becomes the associated complexstructure to ω ⊖ given by J ⊖ = J ⊕ J T , where J i , are the associated complexstructures to ω i , i = 1 , α = ∗ θ ⊖ on Λ is closed since its differential satisfies dα = ∗ dθ ⊖ = ∗ ω ⊖ ≡ . Applying Poincare’s lemma, α is locally exact on Λ and there (locally) existsa function S : Λ → R defined on Λ such that its differential concides with thepullback of θ ⊖ to Λ, i.e. dS = α = ∗ θ ⊖ . The function S : Λ → R is called a generating function for the symplectic map φ : ( M, ω ) → ( M, ω ). In fact thegenerating function is a function ˆ S : P → R defined on P and the function S isthe composition S ≡ ˆ S ◦ : Λ → R .An embedding of a 2 n -dimensional manifold : Λ ֒ → ( P , ω ⊖ ) is called La-grangian if ∗ ω ⊖ ≡
0. Given a Lagrangian embedding : Λ ֒ → ( P , ω ⊖ ), thereexists an open neighborhood Λ ⊂ U ⊂ P around Λ and a projection π : U → Λ,such that the composition Λ ֒ −→ U π → Λ satisfies π ◦ = id Λ . This fact isjust Weinstein’s theorem saying that U is locally symplectomorphic to an openneighborhood of the zero section in T ∗ Λ. A Liouvillian form θ on ( P , ω ⊖ = dθ )is related to the generating function S : Λ → R by the identity dS = ∗ θ , and itsatisfies ker θ ⊂ ker π ∗ ( dS ), equivalently ker θ ⊂ ∗ ( T Λ). The last relation is allwe need to know to construct symplectic maps from Liouvillian forms.
The method of generating functions uses two different symplectic structures on P , usually denoted by ω ⊖ and ω ⊕ , for working with Lagrangian submanifolds [7,2]. It implicitly uses a twist diffeomorphism known as the canonical isomorphismfor cotangent bundles, relating T ∗ Q × T ∗ Q ∼ = T ∗ ( Q × Q ) . For the construction of symplectic maps, a different twist diffeomorphismis applied solving an alternative Hamiltonian system [3]. This diffeomorphismrelates the product manifold with the double cotangent bundle T ∗ Q × T ∗ Q ∼ = T ∗ ( T ∗ Q ) , and defines a projection by composition T ∗ Q × T ∗ Q → T ∗ ( T ∗ Q ) π T ∗Q → T ∗ Q . The way we select the twist Φ will define a different projection which, by theway, it determines a particular type of generating function.In this paper, we avoid the twist diffeomorphisms and the uncomfortablesituation of working with different symplectomorphic manifolds. Instead of thetwist diffeomorphisms, we consider only the product manifold P , and we definea quaternionic or almost hypercomplex structure on P given by { ( I n , I , J , K} ⊂
End ( T P ) [2]. In local coordinates, we have the matricial representation I = (cid:0) n − I n I n n (cid:1) , J = (cid:16) J n n n J T n (cid:17) and K = (cid:0) n J n J n n (cid:1) , (1)satisfying I = J = K = IJ K = − I n , IJ = K , J K = I , KI = J . (2)We obtain an equivalent framework to the usual one, and it is easy to provethat it just corresponds to a relabeling of coordinates.Let g be the Riemannian structure on P which pointwise corresponds to theEuclidean structure h· , ·i on T x P , x ∈ P and define three symplectic forms by ω I ( · , · ) = g ( · , I· ) , ω J ( · , · ) = g ( · , J · ) and ω K ( · , · ) = g ( · , K· ) , (in particular ω J ≡ ω ⊖ and I ≡ J T n ).Let Λ be a 2 n -dimensional manifold and : Λ ֒ → P an embedding in theproduct manifold P . Consider a tubular neighborhood Λ ⊂ U ⊂ P around Λbeing diffeomorphic to an open neighborhood around the zero section in T ∗ Λsuch that the projection π : U → Λ is well-defined and π ◦ = id Λ .The following result characterizes the submanifolds Λ which are adapted forconstructing non-degenerated local symplectic maps. Theorem 2.1
If the image Λ ⊂ U ⊆ P is a Lagrangian submanifold withrespect to both ω I and ω J then:1. it is a symplectic submanifold with respect to ω K ,2. the kernel of the projection π : U → Λ defines a local symplectic map bythe equation π ∗ ( J ( v )) = π ∗ ( I ( v )) = 0 , x ∈ Λ , v ∈ T x Λ . (3) We use I = J T n in accordance to complex geometry. See the discussion in [7, Rmk. 3.1.6]. Note the similarity of the conditions on Λ ⊂ P with those for Special Lagrangian sub-manifolds in K¨ahler or Calabi-Yau manifolds. See in particular [5, Sec 8.1.1]. uch a map corresponds to the Cayley transformation of some Hamiltonianmatrix H ∈ End ( T M ) .Proof of 1. For every vector v ∈ T x Λ which is tangent to the submanifold Λ,the vectors J ( v ) and I ( v ) are normals to Λ since it is Lagrangian for ω I and ω J , i.e. J ( v ) , I ( v ) ∈ ( T x Λ) ⊥ . In the same way, for every u ∈ ( T x Λ) ⊥ we have J ( u ) , I ( u ) ∈ T x Λ and consequently
I ◦ J ( v ) = −J ◦ I ( v ) = K ( v ) ∈ T x Λ . (4)This shows that T x Λ and ( T x Λ) ⊥ are invariant under the action of K whichimplies that Λ ⊂ P is a symplectic submanifold for ω K . Moreover, given theprojection π : P → Λ we have π ∗ ( J ( v )) = π ∗ ( I ( v )) = 0. This is just the factthat ker π ≡ ( T Λ) ⊥ . (cid:3) For proving the point we need local coordinates and some additionalelements. In fact, the proof is to explain how we can construct symplectic mapsusing Liouvillian forms. This is the subject of the following section. Consider the same hypotheses of Theorem 2.1. For constructing symplecticmaps using Liouvillian forms, consider an element v ∈ T x Λ ⊂ T x P , and searchfor primitive forms θ I and θ J such that v ∈ ker θ J ∩ ker θ I . Since Λ is Lagrangianfor ω I and ω J , then J ( v ) , I ( v ) ∈ ( T x Λ) ⊥ . Since ( T x Λ) ⊥ ≡ ker π , this implies π ∗ I ( v ) = π ∗ J ( v ) = 0. We will prove, in a contructive way, that the symplecticmap is the solution of the equation π ∗ I ( v ) = 0. For free, we obtain thatsolving for a set of coordinates of one of the factor manifolds, gives the Cayley’stransformation for some Hamiltonian matrix H .Consider a local vector field Z around Λ being Liouville for both ω I and ω J .We obtain the Liouvillian forms θ I = i Z ω I and θ J = i Z ω J by contraction. Let x ∈ Λ ⊂ P be a point and v := Z ( x ) ∈ T x Λ the element of Z on T x Λ. Then v ∈ ker θ I ∩ ker θ J by construction. We will construct a Liouville vector field Z being suitable for constructing symplectic maps. Lemma 2.2
Let { x i } ni =1 be local coordinates on P . The “expanding” or “Euler”vector field Z ∈ Γ( T P ) , given in these coordinates by Z = P i x i ∂∂x i , isLiouville for all the three symplectic forms ω I , ω J and ω K .Proof. A direct verification shows that d ◦ i X ( ω C ) = ω C , C ∈ {I , J , K} , and θ C = i X ω C is a Liouvillian form for ( P , ω C ). (cid:3) Remark 1 The expanding vector field Z is a degenerated case which correspondsto the identity map. In [4] it is proved that the symplectic integrator contructedwith the expanding vector field corresponds to the mid point rule.
4e proceed by looking for Liouville vector fields Z ∈ ker θ I ∩ ker θ J closeto the expanding vector field. This is achieved by the addition of a componentto the vector fields which is Hamiltonian with respect to ω I and ω J . We solvethis problem in the linear case.A linear vector field Y = P i A ij x j ∂∂x i on P is Hamiltonian for ω C , C ∈{I , J , K} , if A = ( A ij ) is a Hamiltonian matrix for the corresponding complexstructure, i.e. if A T C + CA = 0 holds. Lemma 2.3
Let
S, R ∈ M n × n ( R ) be a symmetric and a Hamiltonian ma-trix respectively, for the n -dimensional symplectic manifold ( M, ω ) . Then thematrix A ∈ M n × n ( R ) given by A = (cid:16) R S − JSJ − R T (cid:17) . (5) is Hamiltonian for ( P , ω I ) and ( P , ω J ) .Proof. The matrix A is Hamiltonian for both ω I and ω J if it satisfies simulta-neously: i) A T I + I A = 0 and ii) A T J + J A = 0.Consider the matrix A = (cid:0) A A A A (cid:1) and solving equation A T I + I A = 0 givesthe conditions A = A T , A = A T and A = − A T . It means i ) requires that A and A be symmetric and it relates A with A . On the other hand the equation A T J + J A = 0 gives the conditions A T J + JA = 0 and A = − JA T J . Itmeans ii ) requires that A and A be Hamiltonian and it relates A and A . Ifwe denote R = A and S = A then A = − JS T J and A = − R T . Finally, R must be Hamiltonian for ω on M , and S symmetric. This gives A by expression(5) which proves the lemma. (cid:3) If we consider that A is not Hamiltonian for ω K then A T K + K A = 0. Thisproduces the additional conditions R = − R T or S = S T . Since S = S T isalready a constraint from Lemma 2.3, then R cannot be antisymmetric. Inparticular, for R a symmetric, Hamiltonian matrix for ( M, ω ) this problem hassolutions.For the following result, we need local coordinates for each one of the fac-tors in the product manifold (cid:0) { x i } ni =1 , { X i } ni =1 (cid:1) ∈ P = M × M . In thesecoordinates, the pointwise element v = Z ( x, X ) is expressed in matricial formby v = (cid:16) I + R S − JSJ I − R T (cid:17) (cid:18) xX (cid:19) . (6)We are in measure of proving second part of Theorem 2.1. Proof of 2. [Theorem 2.1] Consider the vector v = Z ( x, X ) given in matricialform by (6). This vector is tangent to Λ by hypothesis and it is in the kernelof θ I and θ J by construction. Since Λ ⊂ P is Lagrangian with respect to ω J then J T ( v ) belongs to the normal bundle ( T ( x,X ) Λ) ⊥ . Applying the complexstructure K we have ( K ◦ J T )( v ) = I ( v ) ∈ ( T ( x,X ) Λ) ⊥ , with expression I ( v ) = (cid:18) − JSJ I − R T − I − R − S (cid:19) (cid:18) xX (cid:19) . π ∗ ( I ( v )) = 0 in these local coordinates becomes (cid:2) − JSJ ( x ) + ( I − R T )( X ) (cid:3) + [ − ( I + R )( x ) − S ( X )] = 0 . Rearranging we obtain the matricial equation (cid:2) I − ( R T + S ) (cid:3) X = [ I + ( R + JSJ )] x. Solving for X is possible if R T + S is a non-exceptional matrix. We considerthe case where R = R T and S = JSJ , it means both matrices are symmetricand Hamiltonian. Consequently, H := R T + S = R + JSJ is well-defined andit is a non-exceptional, Hamiltonian matrix for (
M, ω ). We solve for X and weobtain X = ( I − H ) − ( I + H ) x. The
Cayley’s transformation [8] assures that the matrix S = ( I − H ) − ( I + H )is symplectic if, and only if H is Hamiltonian, and consequently the map x ( I − H ) − ( I + H ) x is a linear symplectic transformation. (cid:3) References [1] V.I. Arnold.
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