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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, it was shown that the symplectic reduction of a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way and that any special manifold or orbifold with such a connection is locally equivalent to one of these symplectic reductions.
Abstract: By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.
21 citations
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TL;DR: The Schroedinger equation is a lift of Newton's law of motion on the space of probability measures, where derivatives are taken wrt the Wasserstein Riemannian metric as mentioned in this paper, where the potential is the sum of the total classical potential energy of the extended system and its Fisher information.
Abstract: We show that the Schroedinger equation is a lift of Newton's law of motion on the space of probability measures, where derivatives are taken wrt the Wasserstein Riemannian metric Here the potential is the sum of the total classical potential energy of the extended system and its Fisher information The precise relation is established via a well known ('Madelung') transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures
21 citations
TL;DR: In this article, a construction of multidimensional parametric Yang-Baxter maps is presented, where the corresponding Lax matrices are the symplectic leaves of first degree matrix polynomials equipped with the Sklyanin bracket.
Abstract: A construction of multidimensional parametric Yang-Baxter maps is presented. The corresponding Lax matrices are the symplectic leaves of first degree matrix polynomials equipped with the Sklyanin bracket. These maps are symplectic with respect to the reduced symplectic structure on these leaves and provide examples of integrable mappings. An interesting family of quadrirational symplectic YB maps on $\mathbb{C}^4 \times \mathbb{C}^4$ with $3\times 3$ Lax matrices is also presented.
21 citations
TL;DR: In this paper, a simplification of Donaldson's arguments for the construction of symplectic hypersurfaces or Lefschetz pencils is presented, which makes it possible to avoid any reference to Yomdin's work on the complexity of real algebraic sets.
Abstract: We describe a simplification of Donaldson's arguments for the construction of symplectic hypersurfaces or Lefschetz pencils that makes it possible to avoid any reference to Yomdin's work on the complexity of real algebraic sets.
21 citations
TL;DR: A unified approach to obtain symplectic integrators of arbitrarily high order from Lie group integrators on a Lie group G from Runge–Kutta–Munthe-Kaas methods and Crouch–Grossman methods is presented.
Abstract: In this article, a unified approach to obtain symplectic integrators on $$T^{*}G$$T?G from Lie group integrators on a Lie group $$G$$G is presented. The approach is worked out in detail for symplectic integrators based on Runge---Kutta---Munthe-Kaas methods and Crouch---Grossman methods. These methods can be interpreted as symplectic partitioned Runge---Kutta methods extended to the Lie group setting in two different ways. In both cases, we show that it is possible to obtain symplectic integrators of arbitrarily high order by this approach.
20 citations