Showing papers on "Symplectic manifold published in 1986"
Journal Article•
298 citations
TL;DR: In this article, the Lagrangian and Legendrian knots of contact manifolds have been studied and generalized to manifold embeddings, and two theorems of Givental' on Lagrangians have been proved.
Abstract: CONTENTSIntroduction § 1. Is there such a thing as symplectic topology? § 2. Generalizations of the geometric theorem of Poincare § 3. Hyperbolic Morse theory § 4. Intersections of Lagrangian manifolds § 5. Legendrian submanifolds of contact manifolds § 6. Lagrangian and Legendrian knots § 7. Two theorems of Givental' on Lagrangian embeddings § 8. Odd-dimensional analogues § 9. Optical Lagrangian manifoldsReferences
177 citations
TL;DR: In this article, the authors studied the use of unitary unitary similarity transformations in the QR decomposition, where S is either unitary or unitary symplectic, respectively.
110 citations
Book•
01 Jan 1986
TL;DR: In this article, the authors define an isomorphism between the tangent and cotangent bundles of a manifold and define a 2n-dimensional manifold called a symplectic manifold.
Abstract: 1. Basic definitions, examples, and problems. 1.1. Symplectic forms and manifolds. An exterior differential 2-form w o n a smooth manifold V is called nonsingular if the associated homomorphism between the tangent and cotangent bundles of V, denoted Jw : T(V) —• T*(V) and defined by / w ( 0 = w(r, •), is an isomorphism. In this case, the dimension of V is necessarily even (we assume that V is finite-dimensional and all connected components of V have the same dimension), say dimV = m = 2n, and the exterior power oj (which is a top-dimensional form on V) does not vanish on V. Conversely, if uj does not vanish, then UJ is nonsingular. For example, every (oriented) area form on a surface is nonsingular. A form u) on V is called symplectic if it is nonsingular and closed, that is, du) = 0. Then (V, UJ) is called a symplectic manifold. EXAMPLES. Every surface with an area form is a symplectic manifold. If (Vi>Ui) are such surfaces for i = 1 , . . . , n, then the Cartesian product (V^) = (Vi X V2 X • • • X Vn, u)i © u)2 © • • • © wn) is a 2n-dimensional symplectic manifold. The symplectic area w(S) = fs u of every surface S in this V equals the sum of the (oriented!) areas of the projections S —• Vi. An important special case is the symplectic space (R n ,o; = ]C"=i ^ % Adj/i); that is, the sum of n copies of the (z,2/)-plane R 2 with the usual area form dx A dy. A less obvious example is the complex projective space CP which admits a unique (up to a scalar multiple) 2-form UJ which is invariant under the action of the unitary group U(n + 1) on CP. This form is (easily seen to be) symplectic, and the symplectic area of every surface S C CP equals the average number of the intersection points (counted with algebraic multiplicity) of S with hyperplanes p C C P n , u(s) = J#(snp)dPi
69 citations
64 citations
TL;DR: In this article, the authors studied the Weyl transforms of the tempered distributions on the phase space and that of the star-exponentials which gave the spectrum in this process of quantization.
54 citations
TL;DR: In this article, a decomposition of the nuclear Hubert space into collective subspaces and symplectic shells is proposed, and the resultant symplectic shell model is shown to provide a basis for microscopic calculations which are immediately interpretable in collective-model terms.
43 citations
30 citations
TL;DR: The geometry of the bundle T (M, ω) of the compatible complex structures of the tangent spaces of an almost-symmetric manifold is described in this article.
26 citations
01 Jan 1986
TL;DR: In this article, the authors present a brief survey of considerations and results of a study under-taken by the author and his group, on the links between the Hamiltonian formalism and the numerical methods for solving dynamical problems expressed in the form of the canonical system of diierential equations.
Abstract: It is an honor and a pleasure for me to present the inaugural talk at the Tenth International Conference on Numerical Methods in Fluid Dynamics in Beijing. We present a brief survey of considerations and results of a study 1, 2, 3, 4, 6], under-taken by the author and his group, on the links between the Hamiltonian formalism and the numerical methods for solving dynamical problems expressed in the form of the canonical system of diierential equations dp i The canonical system (1.1) with remarkable elegance and symmetry was introduced by Hamilton as a general mathematical scheme, rst for problems of geometrical optics in 1824, then for conservative dynamical problems in 1834. The approach was followed and developed further by Jacobi into a well-established mathematical formalism for analytical dynamics, which is an alternative of, and equivalent to, the Newtonian and Lagrangian formalisms. The geometrization of the Hamiltonian formalism was undertaken by Poincare in 1890's and by Cartan, Birkhoo, Weyl, Siegel, etc., in the 20th century; this gave rise a new discipline, called symplectic geometry, which serves as the mathematical foundation of the Hamiltonian formalism. It is known that, Hamiltonian formalism, apart from its classical links with analytical mechanics, geometrical optics, calculus of variations and non-linear PDE of rst order, has inherent connections also with unitary representations of Lie groups, geometric quantization, pseudo-diierential and Fourier integral operators, classiication of singularities, integrability of non-linear evolution equations, optimal control theory, etc.. It is also under extension to innnite dimensions for various eld theories, including uid dynamics, elasticity, elec-trodynamics, plasma physics, relativity, etc.. Now it is almost certain that all real physical
20 citations
TL;DR: In this paper, a structure theorem for certain kinds of symplectic manifold with a Lagrangian fibration is presented, and the class of cotangent bundles is characterized up to the appropriat equivalence, as the type of manifold considered in the theorem for which a certain cohomology class vanishes.
Abstract: A structure theorem is presented for certain kinds of symplectic manifold with a Lagrangian fibration. As a corollary, the class of cotangent bundles is characterized up to the appropriat equivalence, as the type of symplectic manifold considered in the theorem for which in addition, a certain cohomology class vanishes. These results and techniques are then applied to two situations in classical mechanics where symplectic manifolds foliated by Lagrangian submanifolds arise, namely, the Legendre transformation and Hamilton-Jacobi theory.
01 Jan 1986
TL;DR: In this paper, a construction of symplectic Clifford algebras is given, based on an algebraic process analogous to the one used in orthogonal Clifford algesbras.
Abstract: We give a construction of symplectic Clifford algebras according to an algebraic process analogous to the one used in orthogonal Clifford algebras. We define Clifford and spinors groups and symplectic spinors. We develop two applications: first a geometric approach to the Fourier transform, second a deformation theory for the algebras associated with a symplectic manifold.
TL;DR: In this article, the authors determine the symplectic Hamiltonian structure associated with the nonlinear evolution equations obtained from two new isospectral problems and follow the method of variation with respect to the field variables.
Abstract: We determine the symplectic Hamiltonian structure associated with the nonlinear evolution equations obtained from two new isospectral problems. We follow the method of variation with respect to the field variables. An explicit example is given to demonstrate the new class of equations that are generated.
TL;DR: In this paper, the symmetric structure of a non-Abelian Higgs model on the lattice is calculated in terms of gauge invariants, and the symmetry of the model is analyzed.
TL;DR: The Grassmann manifold is a submanifold of the projective space of the full many-fermion Hilbert space as discussed by the authors, and the Grassmann variety is the minimal coadjoint orbit.
Abstract: Mean field theory is given a geometrical interpretation as a Hamiltonian dynamical system. The Hartree-Fock phase space is the Grassmann manifold, a symplectic submanifold of the projective space of the full many-fermion Hilbert space. The integral curves of the Hartree-Fock vector field are the time-dependent Hartree-Fock solutions, while the critical points of the energy function are the time-independent states. The mean field theory is generalized beyond determinants to coadjoint orbit spaces of the unitary group; the Grassmann variety is the minimal coadjoint orbit.
01 Jan 1986
TL;DR: Singularities and linearization of the fundamental vector field on an exact symplectic manifold have been applied to the characterization of the cotangent structures as discussed by the authors and linearisation du champ fondamental on a variete symplectique exact.
Abstract: Singularities and linearization of the fundamental vector field on an exact symplectic manifold. Application to the characterization of the cotangent structures Singularites et linearisation du champ fondamental sur une variete symplectique exacte. Application a la caracterisation des structures cotangentes
01 Jan 1986
TL;DR: Many questions in singularity theory (for instance, the classification of the singularities of caustics and wave fronts, and also the investigation of the various singularities in optimization and variational calculus problems) become understandable only within the framework of the geometry of symplectic and contact manifolds, which is refreshingly unlike the usual geometries of Euclid, Lobachevskij and Riemann as discussed by the authors.
Abstract: Many questions in singularity theory (for instance, the classification of the singularities of caustics and wave fronts, and also the investigation of the various singularities in optimization and variational calculus problems) become understandable only within the framework of the geometry of symplectic and contact manifolds, which is refreshingly unlike the usual geometries of Euclid, Lobachevskij and Riemann.
01 Jan 1986
TL;DR: In this article, a procedure for the matrix representation of Lie algebras introduced by Gruber and Klimyk can be applied to the symplectic model of the nucleus.
Abstract: The purpose of this note is to show that a procedure for the matrix representation of Lie algebras introduced by Gruber and Klimyk can be applied to symplectic ones and, when combined with a Dyson type boson realization of these algebras, it provides a convenient technique for the matrix representation of their generators We illustrate the procedure for the case of sp(4,R) and indicate how its extension to sp(6,R) can be useful for applications to the symplectic model of the nucleus