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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.
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TL;DR: In this paper, generalized Calabi-Yau structures are studied in the case of K3 surfaces and the moduli space admits a canonical symplectic form with respect to which the modulus space of symplectic structures is Lagrangian.
Abstract: Generalized Calabi–Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi–Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period map and a weak form of the Global Torelli theorem.
72 citations
TL;DR: In this paper, it was shown that Kahan's discretization of quadratic vector fields is equivalent to a Runge-Kutta method, which produces large classes of integrable rational mappings in two and three dimensions.
Abstract: We show that Kahan's discretization of quadratic vector fields is equivalent to a Runge–Kutta method. When the vector field is Hamiltonian on either a symplectic vector space or a Poisson vector space with constant Poisson structure, the map determined by this discretization has a conserved modified Hamiltonian and an invariant measure, a combination previously unknown amongst Runge–Kutta methods applied to nonlinear vector fields. This produces large classes of integrable rational mappings in two and three dimensions, explaining some of the integrable cases that were previously known.
71 citations
TL;DR: The relation between the symplectic structures on the canonical and radiative phase spaces of general relativity has been investigated in this article, where the authors show that the relation between these structures can be expressed as follows:
Abstract: The relation between the symplectic structures on the canonical and radiative phase spaces of general relativity is exhibited.
70 citations
TL;DR: In this paper, the authors use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space.
Abstract: We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting linearly on the ambient space and consequently preserve the corresponding momentum. These results provide an elementary construction of symplectic integrators for Lie-Poisson systems and other Hamiltonian systems with symmetry. The methods are illustrated on the free rigid body, the heavy top, and the double spherical pendulum.
70 citations
70 citations