Showing papers on "Symplectic vector space published in 1989"

Symplectic fixed points and holomorphic spheres

Abstract: LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It is shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we show that if the fixed point set of an exact diffeomorphism onP is nondegenerate, then it satisfies the Morse inequalities onP. We also discuss fixed point estimates for general exact diffeomorphisms.

Symplectic topology and Hamiltonian dynamics

Abstract: On etudie des applications symplectiques non lineaires. Capacites symplectiques. Construction d'une capacite symplectique. Problemes de plongement. Problemes de rigidite

Periodic maps in symplectic topology

Abstract: Symplectic topology arose with [I], where a conjecture on fixed points of symplectomorphisms generalizing the "Poincare geometric theorem" [2, 3] was formulated. This conjecture of Arnol'd gives as a lower bound for the number of fixed points of symplectomorphis which is a homologicai identity of a closed symplectic manifold* the minimal number of critical points of a function on this manifold. The present paper is basically devoted to a discussion of the following conjecture.

R 2n is a universal symplectic manifold for reduction

Abstract: We show that R 2n with its standard symplectic structure is ‘universal’ in that, subject to a mild topological restriction, essentially all symplectic manifolds can be obtained from it by reduction.

Multimode two-photon coherent states and unitary representation of the symplectic group

Abstract: Pure coherent squeezed states of many degrees of freedom are analyzed via a fluctuation matrix description and a symplectic group description.

The complete classification of generalized homogeneous symplectic manifolds

Abstract: The classification scheme for homogeneous symplectic manifolds is completed and generalized including the case when the symmetries are symplectic only up to a factor and the group is nonconnected. This improved classification also describes the possible coverings in terms of classes of some discrete subgroups. As an application, the possibility of mixed symplectic and antisymplectic symmetries is studied for some physical groups: rotations in three dimensions, the Poincare group and the Galilei group (including inversions). Some new possibilities (with compactified energy), not appearing previously in the literature, have been found even for the connected Galilei group.

Berry's phase and the planar three-body problem

Abstract: Berry's angular 2-form in seen as a correction to the symplectic structure in a separation-of-variables-type scheme, where the variables are canonically non-separable. This view is applied to the planar three-body problem where the rotational and vibrational motions are not separable. It is shown that the corrected symplectic structure gives the correct quantisation.

Symplectic structure of Yang-Mills theory in BRST formalism

Abstract: We construct a symplectic form for classical Yang-Mills theory, which is manifestly covariant and Becchi-Rouet-Stora-Tyutin invariant. This suggests an approach to eliminate the degeneracy that exists in the initially given two-form for the theory.

Symplectic complex bundles over real algebraic four-folds

Abstract: Let A\" be a compact affine real algebraic variety of dimension 4. We compute the Witt group of symplectic bilinear forms over the ring of regular functions from X to C. The Witt group is expressed in terms of some subgroups of the cohomology groups H(X, Z) for k = 1,2. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 57 R 22, 19 G 12.