Topic
Symplectic manifold
About: Symplectic manifold is a research topic. Over the lifetime, 4872 publications have been published within this topic receiving 128908 citations.
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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.
2,482 citations
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TL;DR: For Hamiltonian systems of the form H = T(p)+V(q) a method was shown to construct explicit and time reversible symplectic integrators of higher order as discussed by the authors.
2,080 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
1,928 citations
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TL;DR: In this paper, it was shown that compatible symplectic structures lead in a natural way to hereditary symmetries, and that a hereditary symmetry is an operator-valued function which immediately yields a hierarchy of evolution equations, each having infinitely many commuting symmetry all generated by this hereditary symmetry.
1,651 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
1,556 citations