Jędrzej Śniatycki

Reduction and quantization for singular momentum mappings

When a Hamiltonian action of Lie group on a symplectic manifold has a singular momentum mapping, the reduced manifold may not exist. Nevertheless, we may always construct a Poisson algebra which corresponds to the functions on the reduced manifold in the regular case. The ideas of geometric quantization are extended to Poisson algebras, and it is shown in an example that quantization may be carried out before or after reduction, with isomorphic results.

On the quantization of presymplectic dynamical systems via coisotropic imbeddings

We study certain aspects of the problem of quantizing a presymplectic dynamical system. Such a system is quantized by imbedding the presymplectic manifoldM under consideration into a symplectic manifoldX and then geometrically quantizing the latter. It is known that this procedure will yield consistent results provided the imbeddingM →X is coisotropic; we show that such imbeddings always exist and are “locally unique.” Furthermore, we investigate the extent to which the resulting quantum dynamics is independent of the choice of coisotropic imbedding; that is, we examine to what extent the presymplectic phase spaceM determines the quantum representation space and the quantization of observables. The quantization is carried out within the geometric quantization framework of Kostant and Souriau.

Geometry of nonholonomically constrained systems

Nonholonomically Constrained Motions Group Actions and Orbit Spaces Symmetry and Reduction Reconstruction, Relative Equilibria and Periodic Orbits Caratheodorys Sleigh Convex Rolling Rigid Body The Rolling Disk.

Closed forms on symplectic fibre bundles

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.

Differential Structure of Orbit Spaces

We present a new approach to singular reduction of Hamiltonian systems with symmetries. The tools we use are the category of differential spaces of Sikorski and the Stefan-Sussmann theorem. The former is applied to analyze the differential structure of the spaces involved and the latter is used to prove that some of these spaces are smooth manifolds. Our main result is the identification of accessible sets of the generalized distribution spanned by the Hamiltonian vector fields of invariant functions with singular reduced spaces. We are also able to describe the differential structure of a singular reduced space corresponding to a coadjoint orbit which need not be locally closed.

The existence and uniqueness of solutions of Yang-Mills equations with bag boundary conditions

The Cauchy problem for the Yang-Mills equations in the Coulomb gauge is studied on a compact, connected and simply connected Riemannian manifold with boundary. An existence and uniqueness theorem for the evolution equations is proven for fields with Cauchy data in an appropriate Sobolev space. The proof is based the Hodge decomposition of the Yang-Mills fields and the theory of non-linear semigroups.

On Action-Angle Variables

In this article we give a new proof of the construction of action-angle variables for completely integrable Hamiltonian systems. We also provide a proof of a theorem originally due to EI~RESMANN [5, 17] (which is of interest in its own right) in order to understand the different roles that the differential topology and the symplectic geometry play. Various notes and comments may be found at the end of the proof of the theorem. The chief merit of our approach is the ability to deal with the actions directly as a momentum map. For the sake of definiteness we are working in the category of C ~ functions and mappings. The results are also valid in the analytic and C k categories; for the latter a little bookkeeping on the number of derivatives is needed.

Fermat's principle in elastodynamics

In general anisotropic inhomogeneous elasticity, disturbances propagate along rays which are neither straight nor perpendicular to the wave fronts, but, as in optics, they are still characterized by their rendering the time of travel between two points stationary.

Yang-Mills and Dirac Fields in a Bag, Existence and Uniqueness Theorems

The Cauchy problem for the Yang-Mills-Dirac system with minimal coupling is studied under the MIT quark bag boundary conditions. An existence and uniqueness theorem for the free Dirac equation is proven under that boundary condition. The existence and uniqueness of the classical time evolution of the Yang-Mills-Dirac system in a bag is shown. To ensure sufficient differentiability of the fields we need additional boundary conditions. In the proof we use the Hodge decomposition of Yang-Mills fields and the theory of non-linear semigroups.

Yang-Mills and Dirac fields in a bag, constraints and reduction

The structure of the constraint set in the Yang-Mills-Dirac theory in a contractible bounded domain is analysed under the bag boundary conditions. The gauge symmetry group is identified, and it is proved that its action on the phase space is proper and admits slices. The reduced phase space is shown to be the union of symplectic manifolds, each of which corresponds to a definite mode of symmetry breaking.