Showing papers on "Symplectic vector space published in 1984"
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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
758 citations
TL;DR: In this article, the authors define the group of diffeomorphisms of a connected smooth manifold X which preserve the symplectic form o2, topologized as usual as the direct limit of the subgroups Diff~X of all diffEomorphisms with support in the fixed compact set K. They give rise to a global homomorphism s o from the universal cover of Difffo0X to HI(X; F).
Abstract: Let X be a connected smooth manifold without boundary. We will write Difff, oX for the group of all compactly supported diffeomorphisms of X which preserve the symplectic form o2, topologized as usual as the direct limit of the subgroups Diff~X of all diffeomorphisms with support in the fixed compact set K. Let Diff, ooX be its identity component. The flux is locally a homomorphism from a neighbourhood of the identity in Diffo, oX to H~(X;R). (See Calabi [6].) It gives rise to a global homomorphism s o from the universal cover of Difffo0X to HI(X; F,), and hence to a homomorphism
38 citations
29 citations
18 citations
TL;DR: The weak orthogonal and weak symplectic groups were shown to be bireflectional under the assumption that (V,f) is regular and char K = 2 as mentioned in this paper.
11 citations
8 citations
TL;DR: In this paper, the dynamics of autonomous non-conservative systems are studied in terms of Lagrangian submanifolds of a special symplectic manifold and the transition between the two descriptions is established by means of the generating function of a symplectic relation.
Abstract: The dynamics of autonomous nonconservative systems is studied in terms of Lagrangian submanifolds of a special symplectic manifold. Both the Hamiltonian and Lagrangian description are taken into consideration and the transition between the two descriptions is established by means of the generating function of a symplectic relation.
8 citations
TL;DR: In this article, the authors studied the correspondence between generating functions of linear symplectic relations and kernels of integral operators of quantum theories, and showed that these kernels can be obtained from linear nonlinear relations generated by quadratic functions.
5 citations
01 Jan 1984
TL;DR: In this paper, the authors established that the concept of real Heisenberg nilpotent group A(ℝ) lies at the foundations of both signal theory and quantum mechanics.
Abstract: In radar analysis there exists an analogue of the Heisenberg uncertainty principle of quantum mechanics. Quantum mechanics stands here for the quantum-mechanical description, at a given instant of time, of a non-relativistic particle. These uncertainty principles suggest that there should exist a common mathematical structure behind both quantum mechanics and the theory of signals. It is the aim of the present article to establish that the concept of real Heisenberg nilpotent group A(ℝ) lies at the foundations of both of these fields. The crucial point is to endow the time-frequency plane ℝ ⊕ ℝ with the structure of a two dimensional real symplectic vector space so that it gets symplectomorphic to the tangent plane to the “Schrodinger coadjoint orbit” at the point 1 in the Kirillov orbit picture for the unitary dual of A(ℝ). Various applications of this geometric relationship between signal theory and nil potent harmonic analysis are pointed out.
4 citations
TL;DR: In this paper, a new set of normal forms for the elements of the real symplectic algebra is proposed to compute their centralizers explicitly, which are then used to identify real linear Hamiltonian systems.
Abstract: Momentum mapping of an autonomous, real linear Hamiltonian system is determined by its set of quadratic integrals. Such a system can be identified with an element of the real symplectic algebra and its quadratic integrals correspond to the centralizer of this element inside the symplectic algebra. In this paper, using a new set of normal forms for the elements of the real symplectic algebra, we compute their centralizers explicitly.
TL;DR: In this paper, the authors studied the smooth actions of a symplectic group on a closed orientable manifold X such that X~Pa(H)xPb(H), under the conditions: a-\\-b ^2n-2 and n^7.
Abstract: We shall study smooth actions of symplectic group Sp(ri) on a closed orientable manifold X such that X~Pa(H)xPb(H), under the conditions: a-\\-b ^2n—2 and n^7. Our result is stated in §2 and proved in §5. Typical examples are given in §1. Similar result on smooth actions of special unitary group SU(n) on a closed orientable manifold X such that X~Pa(C)xPb(C) is stated in the final section. Throughout this paper, let H*( ) denote the singular cohomology theory with rational coefficients, and let Pn(H), Pn(C) and Pn(R) denote the quaternion, complex and real projective n-space, respectively. By X~X', we mean that H*(X)