Topic

# Symplectic vector space

About: Symplectic vector space is a(n) research topic. Over the lifetime, 2048 publication(s) have been published within this topic receiving 53456 citation(s).

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TL;DR: In this article, the authors define a parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J).

Abstract: Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V which is a smooth proper embedding. A curve is called rational if one can choose S diffeomorphic to the sphere S 2.

2,336 citations

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01 Jan 1995

TL;DR: In this article, the authors present a survey of the history of classical and modern manifold geometry, from classical to modern, including linear and almost complex structures, and the Arnold conjecture of the group of symplectomorphisms.

Abstract: Introduction I. FOUNDATIONS 1. From classical to modern 2. Linear symplectic geometry 3. Symplectic manifolds 4. Almost complex structures II. SYMPLECTIC MANIFOLDS 5. Symplectic group actions 6. Symplectic fibrations 7. Constructing symplectic manifolds III. SYMPLECTOMORPHISMS 8. Area-preserving diffeomorphisms 9. Generating functions 10. The group of symplectomorphisms IV. SYMPLECTIC INVARIANTS 11. The Arnold conjecture 12. Symplectic capacities 13. New directions

1,919 citations

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01 Jan 1984

TL;DR: The geometry of the moment map and motion in a Yang-Mills field and the principle of general covariance have been studied in this paper, where they have been shown to be complete integrability and contractions of symplectic homogeneous spaces.

Abstract: Preface 1 Introduction 2 The geometry of the moment map 3 Motion in a Yang-Mills field and the principle of general covariance 4 Complete integrability 5 Contractions of symplectic homogeneous spaces References Index

1,523 citations

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TL;DR: In this paper, a unified framework for the construction of symplectic manifolds from systems with symmetries is presented, including rotationally invariant systems, the rigid body, fluid flow, and general relativity.

Abstract: We give a unified framework for the construction of symplectic manifolds from systems with symmetries. Several physical and mathematical examples are given; for instance, we obtain Kostant’s result on the symplectic structure of the orbits under the coadjoint representation of a Lie group. The framework also allows us to give a simple derivation of Smale's criterion for relative equilibria. We apply our scheme to various systems, including rotationally invariant systems, the rigid body, fluid flow, and general relativity.

1,389 citations

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01 Jan 1994

TL;DR: Examples of Hamiltonian Systems, symplectic integration, and Numerical Methods: Checking preservation of area: Jacobians, and Necessity of the symplecticness conditions.

Abstract: Hamiltonian Systems. Examples of Hamiltonian Systems. Symplecticness. The solution operator. Preservation of area. Checking preservation of area: Jacobians. Checking preservation of area: differential forms. Symplectic transformations. Conservation of volume. Numerical Methods. Numerical integrators. Stiff problems. Runge-Kutta methods. Partitioned Runge-Kutta methods. Runge-Kutta-Nystrom methods. Composition of methods - adjoints. Order conditions. Order conditions for Runge-Kutta methods. The local error in Runge-Kutta methods. Order conditions for PRK methods. The local error in Partitioned Runge-Kutta methods. Order conditions for Runge-Kutta-Nystrom methods. The local error in Runge-Kutta-Nystrom mehthods. Implementation. Variable step sizes. Embedded pairs. Numerical experience with variable step sizes. Implementing implicit methods. Fourth-order Gauss method. Symplectic integration. Symplectic methods. Symplectic Runge-Kutta methods. Symplectic partitioned Runge-Kutta methods. Symplectic Runge-Kutta-Nystrom methods. Necessity of the symplecticness conditions. Symplectic order conditions. Prelimiaries. Order conditions for symplectic RK methods. Order conditions for symplectic PRK methods. Order conditions for symplectic RKN methods. Homogenous form of the order conditions. Available symplectic methods. Symplecticness of the Gauss methods. Diagonally implicity Runge-Kutta methods. Other symplectic Runge-Kutta methods. Explicit partitioned Runge-Kutta methods. Available symplectic Runge-Kutta-Nystrom methods. Numerical experiments. A comparison of symplectic integrators. Variable step sizes for symplectic methods. Conclusions and recommendations. Properties of symplectic integrators. Backward error interpretation. An alternative approach. Conservation of energy. KAM theory. Generating functions. The concept of generating function. Hamilton-Jacobi equations. Integrators based on generating functions. Generating functions for RK methods. Canonical order theory. Lie formalism. The Poisson bracket. Lie operators and Lie series. The Baker-Campbell-Hausdorff formula. Application to fractional-step methods. Extension to the non-Hamilton case. High-order methods. High-order Lie methods. High-order Runge-Kutta-Nystrom methods. A comparison of order 8 symplectic integrators. Extensions. Partitioned Runge-Kutta methods for nonseparable Hamiltonian systems. Canonical B-series. Conjugate symplectic methods. Trapezoidal rule. Constrained systems. General Poisson structures. Multistep methods. Partial differential equations. Reversable systems. Volume preserving flows.

1,271 citations