Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.

Toward a topological characterization of symplectic manifolds

Abstract: A fibration-like structure called a hyperpencil is defined on a smooth, closed 2n-manifold X, generalizing a linear system of curves on an algebraic variety. A deformation class of hyperpencils is shown to determine an isotopy class of symplectic structures on X. This provides an inverse to Donaldson's program for constructing linear systems on symplectic manifolds. In dimensions ≤ 6, work of Donaldson and Auroux provides hyperpencils on any symplectic manifold, and the author conjectures that this extends to arbitrary dimensions. In dimensions where this holds, the set of deformation classes of hyperpencils canonically maps onto the set of isotopy classes of rational symplectic forms up to positive scale, topologically determining a dense subset of all symplectic forms up to an equivalence relation on hyperpencils. In particular, the existence of a hyperpencil topologically characterizes those manifolds in dimensions ≤ 6 (and perhaps in general) that admit symplectic structures.

A symplectic fixed point theorem for ...Cpn.

Abstract: Two symplectic diffeomorphisms,φ0,φ1 of a symplectic manifold (X, ω) are said to be homologous if there exists a smooth homotopyφ1,t∋[0, 1] of symplectic diffeomorphisms between them such that the timedependent vector fieldξt defined byd/dt(φt-ξtoφt is a globally hamiltonian vector field for allt, i.e. there exists a smooth real-valued timedependent hamiltonian functionh(x, t) onX x [0, 1] such thatξt⌋ω=dht, whereht=h(x,t).

Discrete Routh reduction

Abstract: This paper develops the theory of Abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with Abelian symmetry. The reduction of variational Runge–Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical J2 correction, as well as the double spherical pendulum. The J2 problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a non-trivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the non-canonical nature of the symplectic structure.

Symplectic geometry and topology

Abstract: This is a review of the ideas underlying the application of symplectic geometry to Hamiltonian systems. The paper begins with symplectic manifolds and their Lagrangian submanifolds, covers contact manifolds and their Legendrian submanifolds, and indicates the first steps of symplectic and contact topology.