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A Quaternionic Structure as a Landmark for Symplectic Maps

We use a quaternionic structure on the product of two symplectic manifolds for relating Liouvillian forms with linear symplectic maps obtained by the symplectic Cayley's transformation.


Introduction
One of the main difficulties for constructing symplectic maps by the method of generating functions is the resolution of the Hamilton-Jacobi equation. Instead of solving such an equation, in this paper we consider a local quaternionic structure on the symplectic product manifold and the three different symplectic forms induced by this structure. Symplectic maps are constructed using the primitive Liouvillian forms related to these symplectic forms.

Local symplectic maps from Liouvillian forms
Let (M, ω) be a 2n-dimensional symplectic manifold with symplectic form ω. A symplectomorphism is a diffeomorphism φ : (M, ω) → (M, ω) preserving the symplectic structure φ * ω = ω, where the star stands for the pull-back of differential forms. When the symplectic structure has a global primitive linear form θ, then (M, dθ) is called an exact symplectic manifold. Main representatives are cotangent bundles (T * Q, dα) which possesses a canonical or tautological form α called the Liouville form. We define a Liouvillian form in an exact symplectic manifold, as any representative θ ∈ [α] in Ω 1 (M ), and a Liouville vector 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 at the same time, the points z 0 , z h and the full path of symplectic diffeomorphisms φ t = φ(t) connecting them z h = φ h (z 0 ). Then φ([0, h]) ֒→ B is an embedded segment of curve. In a convex ball, there always exist primitive 1-forms θ by Poincaré's lemma and consequently we can apply this procedure locally on any symplectic manifold.
Consider Γ φ as an embedding  : Λ ֒→ P with (Λ) = Γ φ ⊂ P, and by abuse of notation we identify Λ and its image (Λ). In terms of this embedding we have  * ω ⊖ ≡ 0. In addition to the symplectic form ω ⊖ , for every x ∈ P there exists an induced endomorphism on T x P which becomes the associated complex structure to ω ⊖ given by The linear form α =  * θ ⊖ on Λ is closed since its differential satisfies Applying Poincare's lemma, α is locally exact on Λ and there (locally) exists a function S : Λ → R defined on Λ such that its differential concides with the pullback 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 the generating function is a functionŜ : P → R defined on P and the function S is the composition S ≡Ŝ •  : Λ → R. An embedding of a 2n-dimensional manifold  : Λ ֒→ (P, ω ⊖ ) is called Lagrangian if  * ω ⊖ ≡ 0. Given a Lagrangian embedding  : Λ ֒→ (P, ω ⊖ ), there exists an open neighborhood Λ ⊂ U ⊂ P around Λ and a projection π : U → Λ, such that the composition Λ This fact is just Weinstein's theorem saying that U is locally symplectomorphic to an open neighborhood 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 it satisfies ker θ ⊂ ker π * (dS), equivalently ker θ ⊂  * (T Λ). The last relation is all we need to know to construct symplectic maps from Liouvillian forms.

A quaternionic structure on the product manifold
The method of generating functions uses two different symplectic structures on P, usually denoted by ω ⊖ and ω ⊕ , for working with Lagrangian submanifolds [7,3]. It implicitly uses a twist diffeomorphism known as the canonical isomorphism for cotangent bundles, relating For the construction of symplectic maps, a different twist diffeomorphism is applied solving an alternative Hamiltonian system [3]. This diffeomorphism relates the product manifold with the double cotangent bundle T * Q × T * Q ∼ = T * (T * Q), and defines a projection by composition The way we select the twist Φ will define a different projection which, by the way, it determines a particular type of generating function.
In this paper, we avoid the twist diffeomorphisms and the uncomfortable situation of working with different symplectomorphic manifolds. Instead of the twist diffeomorphisms, we consider only the product manifold P, and we define a quaternionic or almost hypercomplex structure on P given by {(I 4n , I, J , K} ⊂ End(T P) [2]. In local coordinates, we have the matricial representation 1 satisfying We obtain an equivalent framework to the usual one, and it is easy to prove that it just corresponds to a relabeling of coordinates. Let g be the Riemannian structure on P which pointwise corresponds to the Euclidean structure ·, · 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 4n ). Let Λ be a 2n-dimensional manifold and  : Λ ֒→ P an embedding in the product manifold P. Consider a tubular neighborhood Λ ⊂ U ⊂ P around Λ being diffeomorphic to an open neighborhood around the zero section in T * Λ such that the projection π : U → Λ is well-defined and π •  = id Λ .
The following result characterizes the submanifolds Λ which are adapted for constructing non-degenerated local symplectic maps.
Theorem 2.1 If the image Λ ⊂ U ⊆ P is a Lagrangian submanifold with respect to both ω I and ω J then: 1. it is a symplectic submanifold 2 with respect to ω K , 2. the kernel of the projection π : U → Λ defines a local symplectic map by the equation

Such a map corresponds to the Cayley transformation of some Hamiltonian matrix 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 This shows that T x Λ and (T x Λ) ⊥ are invariant under the action of K which implies that Λ ⊂ P is a symplectic submanifold for ω K . Moreover, given the projection π : P → Λ we have π * (J (v)) = π * (I(v)) = 0. This is just the fact that ker π ≡ (T Λ) ⊥ . For proving the point 2. we need local coordinates and some additional elements. In fact, the proof is to explain how we can construct symplectic maps using Liouvillian forms. This is the subject of the following section.

Liouvillian forms and symplectic maps
Consider the same hypotheses of Theorem 2.1. For constructing symplectic maps using Liouvillian forms, consider an element v ∈ T x Λ ⊂ T x P, and search for primitive forms θ I and θ J such that v ∈ ker θ J ∩ker θ I . Since Λ is Lagrangian for 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 symplectic map is the solution of the equation π * I(v) = 0. For free, we obtain that solving for a set of coordinates of one of the factor manifolds, gives the Cayley's transformation 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.
i=1 be local coordinates on P. The "expanding" or "Euler" vector field Z 0 ∈ Γ(T P), given in these coordinates by is Liouville for all the three symplectic forms ω I , ω J and ω K .

Proof. A direct verification shows that
and θ C = i X ω C is a Liouvillian form for (P, ω C ).

Remark 1
The expanding vector field Z 0 is a degenerated case which corresponds to the identity map. In [4] it is proved that the symplectic integrator contructed with the expanding vector field corresponds to the mid point rule.
We proceed by looking for Liouville vector fields Z ∈ ker θ I ∩ ker θ J close to the expanding vector field. This is achieved by the addition of a component to the vector fields which is Hamiltonian with respect to ω I and ω J . We solve this problem in the linear case.
A linear vector field Y = i A ij x j ∂ ∂xi on P is Hamiltonian for ω C , C ∈ {I, J , K}, if A = (A ij ) is a Hamiltonian matrix for the corresponding complex structure, i.e. if A T C + CA = 0 holds. Lemma 2.3 Let S, R ∈ M 2n×2n (R) be a symmetric and a Hamiltonian matrix respectively, for the 2n-dimensional symplectic manifold (M, ω). Then the matrix A ∈ M 4n×4n (R) given by is Hamiltonian for (P, ω I ) and (P, ω J ). If we consider that A is not Hamiltonian for ω K then A T K + KA = 0. This produces the additional conditions R = −R T or S = S T . Since S = S T is already a constraint from Lemma 2.3, then R cannot be antisymmetric. In particular, for R a symmetric, Hamiltonian matrix for (M, ω) this problem has solutions.

Proof. The matrix
For the following result, we need local coordinates for each one of the factors in the product manifold In these coordinates, the pointwise element v = Z(x, X) is expressed in matricial form by 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 matricial form by (6). This vector is tangent to Λ by hypothesis and it is in the kernel of 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 complex structure K we have (K • J T )(v) = I(v) ∈ (T (x,X) Λ) ⊥ , with expression Solving for X is possible if R T +S is a non-exceptional matrix. 3 We consider the case where R = R T and S = JSJ, it means both matrices are symmetric and Hamiltonian. Consequently, H := R T + S = R + JSJ is well-defined and it is a non-exceptional, Hamiltonian matrix for (M, ω). We solve for X and we obtain X = (I − H) −1 (I + H)x.