Showing papers on "Symplectic manifold published in 1983"

Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds

Abstract: We prove the existence of star-products and of formal deformations of the Poisson Lie algebra of an arbitrary symplectic manifold Moreover, all the obstructions encountered in the step-wise construction of formal deformations are vanishing

On the symplectic geometry of deformations of a hyperbolic surface

Abstract: Let R be a Riemann surface. In this manuscript we consider a geometry on the moduli space X(R) for R, which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of R. Classically the space of flat metrics for a torus is the locally symmetric space 0(2) \ SL(2; R)/SL(2; Z). We shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative Euler characteristic. The Teichmiiller space T(R), a covering of the moduli space, is a complex Kihler manifold. A KUhler metric for T(R), defined in terms of the Petersson product for automorphic forms, was introduced by Weil, [1]. The Weil-Petersson metric is invariant under the covering transformations and so projects to the moduli space X(R). The metric provides a link between the function theory of R and the geometry of X(R). In the Fenchel-Nielsen manuscript [8] a deformation, based on an amalgamation construction for Fuchsian groups, is introduced. The deformation is defined geometrically by cutting the surface along a simple closed geodesic, rotating one side of the cut relative to the other, and attaching the sides in their new position. The hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface. Choose a free homotopy class [a] on the surface R; then for each marked surface R realize [a] by the closed geodesic aR. The Fenchel-Nielsen deformations for the athen define a 1-parameter group of diffeomorphisms of T(R), whose infinitesimal generator by definition is the Fenchel-Nielsen vector field ta. In [21] the Fenchel-Nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t * was begun. The Fenchel-Nielsen vector fields were found to be related to the geodesic length functions 1*, introduced by Fricke-Klein to provide coordinates for T(R).

Addendum to “on the variation in the cohomology of the symplectic form of the reduced phase space”

Abstract: Here we have used the notation (~rlX) for the inner product of the vector field X with the form a, and ,Ix(m)= (X, J(m)) for the X-component of a. In an earlier paper [4] it was shown that the push forward J,(dm) of the Liouville measure dm on M under the momentum mapping J is a piecewise polynomial measure on !.*. Moreover, in case X has isolated isolated zeros on M an explicit formula for the integral

Singularities of systems of rays

Abstract: CONTENTS Introduction Chapter I. Symplectic geometry § 1. Symplectic manifolds § 2. Submanifolds of a symplectic space § 3. Lagrangian manifolds, bundles, maps, and singularities Chapter II. Applications of the theory of Lagrangian singularities § 4. Oscillatory integral § 5. Integral points § 6. Metamorphoses of caustics Chapter III. Contact geometry § 7. Wave fronts § 8. Singularities of fronts § 9. Metamorphoses of fronts Chapter IV. The convolution of invariants and its generalizations § 10. Vector fields tangent to a front § 11. The linearized convolution of invariants § 12. Period maps and intersection forms Chapter V. Lagrangian and Legendrian topology § 13. Lagrangian and Legendrian cobordisms § 14. Lagrangian and Legendrian characteristic classes Chapter VI. Projections § 15. Singularities of projections of surfaces to a plane § 16. Singularities of projections of complete intersections § 17. The geometry of bifurcation diagrams References

On collective complete integrability according to the method of Thimm

Abstract: Let G be a Lie group acting in Hamiltonian fashion on a symplectic manifold M with moment map Φ:M → g*. A function of the form ƒ∘Φ where ƒ is a function on g* is called ‘collective’. We obtain necessary conditions on the G action for there to exist enough Poisson commuting functions on g* so that the corresponding collective functions on M form a completely integrable system. For the case G = O(n) or U(n) these conditions are sufficient. This explains Thimm's proof [17] of the complete integrability of the geodesic flow on the real and complex grassmanians. We also discuss related questions in the geometry of the moment map.

Closed forms on symplectic fibre bundles

Abstract: A bundle of symplectic manifolds is a differentiable fibre bundle F--~ E-% B whose structure group (not necessarily a Lie group) preserves a symplectic structure on F. The vertical subbundle V=Ker (TTr)_ TE carries a field of bilinear forms which we call the symplectic structure along the fibres and denote by to. Any 2-form 12 on E has a restriction to F; if this restriction is to, we call O an extension of to. In this note, we discuss the problem of finding a closed extension of the symplectic structure along the fibres. This is the first step toward finding a symplectic extension- a problem already considered in special cases in [Th] and [Wn]. The first theorem shows that the existence of a closed extension is a purely topological problem.

Universal unfolding of Hamiltonian systems: From symplectic structure to fiber bundles

Abstract: The problem of the global Lagrangian description of a dynamical system which has a global Hamiltonian description is considered; the motion of a charged particle in the field of a static magnetic monopole is a prototype of such dynamical problems. The symplectic structure on the (cotangent bundle of the) manifold is associated with a closed but not necessarily exact two-form. In conventional language this means that the tensor field inverse to the symplectic metric is divergence free but not necessarily expressible globally as the curl of a vector field; and hence the usual passage to Hamilton's principle and the Lagrangian can be carried out in patches but not necessarily globally. Generalizing techniques developed elsewhere, we show that the fundamental expression for the action integral is like the surface integral of the closed but not necessarily exact antisymmetric tensor field (two-form) rather than like the line integral of a vector field (one-form). This naturally leads to the path-space formalism in which the action functional depends not only on a point in configuration space but also a path from a chosen fixed point terminating in the relevant point in configuration space. We then show how this is a "universal unfolding" and how the topological obstructions in the configuration-space description are circumvented in the path-space description. The path-space description contains redundancies and they can be reduced; locally the new manifold is obtained by associating an angle of rotation [a U(1) fiber] with each configuration-space point, but globally there would be a nontrivial fiber-bundle structure. These questions naturally carry over to the quantum theory of such systems and it is shown how quantization conditions arise. The structure of the quantum-theory formalism for such systems is analyzed. The considerations are applied to the monopole problem where the enlarged space in which the Lagrangian description is possible is explicitly identified as the manifold of the group SU(2).

Fourier Transforms of Orbits of the Coadjoint Representation

Abstract: Let G be a compact Lie group with Lie algebra g. Let 0 ⊂ g* be an orbit of G under the coadjoint representation of maximal dimension 2n. For f ∈ 0, we denote by G(f) the stabilizer of f and t = ℊ(f) the Lie algebra of G(f). Let W be the Weyl group of (g, t). Recall that 0 is a symplectic manifold with a canonical 2-form σ.

L2 estimates for Fourier integral operators with complex phase

Abstract: On relie les techniques de prolongement presque analytiques developpees par A. Melin et J. Sjostrand. Estimations localisees. A*A comme operateur pseudodifferentiel et continuite L 2 . Forme normale pour des relations canoniques lineaires positives. Estimations L 2 precises

Symplectic structure, Lagrangian, and involutiveness of first integrals of the principal chiral field equation

Abstract: We deal with a form of the chiral equation, for which first integrals can be written explicitly. For these equations, we find a symplectic structure, the Lagrangian and first integrals in involution.

Heat, Cold and Geometry

Abstract: Classical and relativistic mechanics can be formulated in terms of symplectic geometry; this formulation leads to a rigorous statement of the principles of statistical mechanics and of thermodynamics.

Deformations of Algebras Associated with a Symplectic Manifold

Abstract: It is possible to give a complete description of Classical Mechanics in terms of symplectic geometry and Poisson brackets. It is the essential of the hamiltonian formalism. In a common program with Flato, D. Sternheimer and J. Vey and other scientists (Fronsdal, Arnal, M. Cahen, S. Gutt, M. de Wilde) we have studied properties and applications of the deformations of the trivial associative algebra and of the Poisson Lie algebra associated with a symplectic manifold. Such deformations give a new approach for Quantum Mechanics; this approach has been developped in other papers ((1),(2)). In this lecture, I will give recent results concerning the existence and the equivalence of associative deformations (or *υ-products).

Normal forms generalizing action-angle coordinates for Hamiltonian actions of Lie groups

Abstract: Let (M, Ω) be a symplectic manifold on which a Lie group G acts by a Hamiltonian action. Under some restrictive assumptions, we show that there exists a symplectic diffeomorphism ψ of a G-invariant open neighbourhood U of a given G-orbit in M, onto an open subset ψ(U) of a vector bundle F*, with base space G. Explicit expressions are given for the symplectic 2-form, for the momentum map and for a Hamiltonian vector field whose Hamiltonian function is G-invariant, on the model symplectic manifold ψ(U).

A lifting theorem for compact symplectic manifolds

Abstract: It is shown that the Lie algebra of globally Hamiltonian vector fields on a compact symplectic manifold can be lifted to a Lie algebra of smooth functions on the manifold under Poisson bracket. This implies that any algebra of symmetries of a classical mechanical system described by such a manifold may be realised as an algebra of observables (smooth functions). Parallels between lifting problems in classical and quantum mechanics are explored.

The symplectic character of the three-dimensional magnetic-binary problem

Abstract: The authors' purpose is to generalize in this paper the symplectic character of the particle motions in the three-dimensional magnetic-binary problem. By adopting the classical electrodynamical formulation they find out the general form of the symplectic matrix as well as some relations between the variations of an orbit, which way be used to check the accuracy of any numerical work related to the orbits investigation.

The Spin Polarizer

Abstract: This communication is to be considered as an addendum to the reference (1) The purpose of that paper was to investigate, within the framework of geometric quantization a la Kostant — Souriau, the new notion of polarizer of a prequantizable symplectic manifold Although that work was mainly concerned with the general scheme of compact semi-simple Lie groups, its ultimate motivation was clearly of physical nature That is why it was there proposed to examine polarizers; a drastic selection of prequantizable coadjoint orbits showed up, which, in fact, might correspond to the actual selection of physically relevant multiplets of hadron spectroscopy The question then arises: must one take polarizers seriously? In order to strenghthen our point of view, we propose here to discuss the properties of polarizers for spinning particles The non relativistic Dirac particle, its Poincare invariant polarizer is explicitely worked out, as well as for the spin 1 relativistic massive particle The associated mixed (real + Kahler) polarization naturally corresponds to the unique Poincare invariant polarization discovered by Souriau and Renouard