We study symplectic surfaces in ruled symplectic 4-manifolds which are disjoint from a given symplectic section. As a consequence, in any symplectic 4-manifold two symplectic surfaces which are C close must be Hamiltonian isotopic.

The connection between closed Newton-Cotes, trigonometrically-fitted differential methods and symplectic integrators is investigated in this paper. It is known from the literature that several one step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. Zhu et al. (1996) presented the well known ope...

The SR algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix B in butterfly form is uniquely determined by 4n− 1 parameters. Using these 4n− 1 parameters, we show how one step of the symplectic SR algorithm for B can be carried out in O(n) arithmetic operations compa...

The SR algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix B in butterfly form is uniquely determined by 4n− 1 parameters. Using these 4n− 1 parameters, we show how one step of the symplectic SR algorithm for B can be carried out in O(n) arithmetic operations compa...

An implicitly restarted symplectic Lanczos method for the symplectic eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. The inherent numerical diiculties of the symplectic Lanczos method are addressed by inexpensive implicit restarts. The method is used to compute some eigenvalues and eigenvectors of large and sparse symplectic matrices.

A Poisson manifold (M2n, π) is b-symplectic if ∧n π is transverse to the zero section. In this paper we apply techniques of Symplectic Topology to address global questions pertaining to b-symplectic manifolds. The main results provide constructions of: b-symplectic submanifolds à la Donaldson, b-symplectic structures on open manifolds by Gromov’s h-principle, and of b-symplectic manifolds with ...

1 Symplectic Manifolds 5 1.1 Symplectic Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Symplectic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Moser’s Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Darboux and Moser Theorems . . . . . . . . . . . . . . . ....

For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic analogues of the twistor operators known from Riemannian spin geometry. We prove...

We describe the structure of the Lie groups endowed with a leftinvariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left canonical action of the group on itself is Hamiltonian. The principal tool used...

We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplectic setting. In particular, we construct scattering-symplectic spheres and sca...