Topic
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 article, the Maslov P-index theory for a symplectic path is defined and various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied.
Abstract: The Maslov P-index theory for a symplectic path is defined. Various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied. As an application, the non-periodic problem for some asymptotically linear Hamiltonian systems is considered.
22 citations
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TL;DR: In this article, a de Rham model for stratified spaces arising from symplectic reduction is introduced, and it turns out that the reduced symplectic form and its powers give rise to well-defined cohomology classes, even on a singular symplectic quotient.
Abstract: We introduce a de Rham model for stratified spaces arising from symplectic reduction. It turns out that the reduced symplectic form and its powers give rise to well-defined cohomology classes, even on a singular symplectic quotient.
22 citations
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TL;DR: In this paper, the existence of normal forms for symplectic maps in R2n is analyzed using a technique based on the generating functions, and explicit algorithms to calculate the analytic expansions of the normal forms and the associated canonical transformation are also provided.
Abstract: The problem of the existence of normal forms for symplectic maps inR
2n is analyzed using a technique based on the generating functions. We discuss extensively the case of a map with an elliptic fixed point: both the resonant and non resonant cases are presented. Explicit algorithms to calculate the analytic expansions of the normal forms and the associated canonical transformation are also provided.
22 citations
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TL;DR: In this paper, the Frobenius distance minimization problem over a non-compact symplectic Lie group Sp(2N,ℝ) has been studied, where the set of critical points has a unique local minimum and saddlepoint submanifolds exhibiting the absence of local suboptima.
Abstract: Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over a noncompact symplectic Lie group Sp(2N,ℝ), i.e., minimizing the Frobenius distance from a target symplectic transformation, which can be used to assess the fidelity function over dynamical transformations in classical mechanics and quantum optics. The topology of the set of critical points is proven to have a unique local minimum and a number of saddlepoint submanifolds, exhibiting the absence of local suboptima that may hinder the search for ultimate optimal solutions. Compared with those of previously studied problems on compact Lie groups, such as the orthogonal and unitary groups, the topology is more complicated due to the significant nonlinearity brought by the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group.
22 citations
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TL;DR: In this article, the construction of the ∗ -product proposed by Fedosov is implemented in terms of the theory of fibre bundles and several properties of the product in the Weyl algebra are proved.
22 citations