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In this paper we define an action of a Lie algebra on a smooth manifold. We get nearly the same results as those for group actions, when the flows of the symmetry vector fields are complete. We show that the orbit space of a Lie algebra action is a differential space. We discuss differential spaces occuring in the reduction of symmetries in integrable Hamiltonian systems.

In this paper, we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This yields an intrinsic view of vector fields and differential forms on the orbit space.

We study geometric quantization of the 1 : 1 in terms of a singular real polarization given by fibres of its energy momentum map.

We construct generalizations of boundary forms of Kupferman, Olami and Segev and of the De Donder form to higher order Lagrangians. In principle, the resulting forms may depend on the coordinate system used. Nevertheless, it leads to many invariant conclusions. In some cases, like second order gravity, the construction leads to a unique De Donder form on the jet bundle that is independent of the choice of coordinates.

For high-order continuum mechanics and classical field theories configurations are modeled as sections of general fiber bundles and generalized velocities are modeled as variations thereof. Smooth stress fields are considered and it is shown that three distinct mathematical stress objects play the roles of the traditional stress tensor of continuum mechanics in Euclidean spaces. These objects are referred to as the variational hyper-stress, the traction hyper-stress and the non-holonomic stress. The properties of these three stress objects and the relations between them are studied.

This paper proves a globalization of a theorem of Horozov (1990). In particular, it determines the inverse of the Horozov map from an explicitly determined simply connected domain in the upper half plane to the set of regular values of the energy–momentum map of the spherical pendulum with the h -axis removed for h ≥ 1 .

We show that derivations of the differential structure of a subcartesian space satisfy the chain rule and have maximal integral curves., Comment: Section on vector fields added

We show that the De Donder form for second order gravity, defined in terms of Ostrogradski's version of the Legendre transformation applied to all independent variables, is globally defined by its local coordinate descriptions. It is a natural differential operator applied to the diffeomorphism invariant Lagrangian of the theory.

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is weakly reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differential forms, that satisfies Smith’s version of de Rham’s theorem. Because the orbit space is a locally closed subcartesian space, it has vector fields and their flows.

The paper considers the formulation of higher-order continuum mechanics on differentiable manifolds devoid of any metric or parallelism structure. For generalized velocities modeled as sections of some vector bundle, a variational kth order hyper-stress is an object that acts on jets of generalized velocities to produce power densities. The traction hyper-stress is introduced as an object that induces hyper-traction fields on the boundaries of subbodies. Additional aspects of multilinear algebra relevant to the analysis of these objects are reviewed., Comment: 23 pages

We show that the space P/G of orbits of a proper action of a Lie group G on a locally compact differential space P is a locally compact differential space with quotient topology. Applying this result to reduction of symmetries of Hamiltonian systems, we prove the reduction by stages theorem

In the framework of geometric quantization we extend the Bohr–Sommerfeld rules to a full quantum theory which resembles the Heisenberg matrix theory. This extension is possible because Bohr–Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1–dimensional harmonic oscillator and the coadjoint orbits of the rotation group.

These lectures notes are meant as an introduction to geometric quantization. In Section 1, I begin with presentation of the historical background of quantum mechanics. I continue with discoveries in the theory of representations of Lie groups, which lead to emergence of geometric quantization as a part of pure mathematics. This presentation is very subjective, flavored by my own understanding of the role of geometric quantization in quantum mechanics and representation theory. Section 2 is devoted to a review of geometry of Hamiltonian systems. Geometric quantization is discussed in the next two sections: prequantization in Section 3 and polarization in Section 4. In particular, I discuss geometric quantization with respect to polarizations given by Kahler structure, cotangent bundle projection and completely integrable system. More advanced topics, like metaplectic structure, pairing of polarizations, and commutation of quantization and reduction, are not included.

We consider a Dirac structure D on a manifold Q invariant under a proper action of a Lie group G on Q. Our aim is to describe the structure of the orbit space D/G in terms of the structure of Q/G.

This paper creates a theory of quantization of singularly reduced systems. We compare our results with those obtained by quantizing algebraically reduced systems. In the case of a Kähler polarization, we show that quantization of a singularly reduced system commutes with reduction, thus generalizing results of Sternberg and Guillemin. We illustrate our theory by treating an example of Arms, Gotay and Jennings where algebraic and singular reduction at the zero level of the momentum mapping differ. In spite of this, their quantizations agree.

We show that there exists a homeomorphism between the closed interval [ 0 , 1 ] ⊂ R and the Koch curve endowed with the subset topology of R 2 . We use this homeomorphism to endow the Koch curve with the structure of a smooth manifold with boundary.

We consider a Hamiltonian action of a connected group G on a symplectic manifold (P, ω) with an equivariant momentum map $$J : P \rightarrow {\mathfrak{g}^{*}}$$ and its quantization in terms of a Kahler polarization which gives rise to a unitary representation $${\mathcal{U}}$$ of G on a Hilbert space $${\mathcal{H}}$$ . If O is a co-adjoint orbit of G quantizable with respect to a Kahler polarization, we describe geometric quantization of algebraic reduction of J −1(O). We show that the space of normalizable states of quantization of algebraic reduction of J −1(O) gives rise to a projection operator onto a closed subspace of $${\mathcal{H}}$$ on which $${\mathcal{U}}$$ is unitarily equivalent to a multiple of the irreducible unitary representation of G corresponding to O. This is a generalization of the results of Guillemin and Sternberg obtained under the assumptions that G and P are compact and that the action of G on P is free. None of these assumptions are needed here.

The aim of this paper is to give a brief description of singular reduction of nonholonomically constrained Hamiltonian systems.

Let be a family of vector fields on a manifold or a subcartesian space spanning a distribution D. We prove that an orbit O of is an integral manifold of D if D is involutive on O and it has constant rank on O. This result implies Frobenius’ theorem, and its various generalizations, on manifolds as well as on subcartesian spaces.

In this paper we give the Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum., Errors in the appendix have been corrected

Based mainly on examples of interest in mechanics, we define the notion of a polite group action. One may view this as not only trying to give a more general notion than properness of a group action, but also to more fully understand the role of invariant functions in describing just about everything of interest in reduction.We show that a polite action of a symmetry group of a dynamical system admits reduction and reconstruction.

The elastica is a curve in $\R^3$ that is stationary under variations of the integral of the square of the curvature. Elastica is viewed as a dynamical system that arises from the second order calculus of variations, and its quantization is discussed., Comment: 55 pages

We investigate four Poisson algebras occurring in the reduction of symmetries of Hamiltonian systems. We show that all four algebras are naturally isomorphic if the action of the symmetry group is free and proper. For a compact group action reduced at the zero level of the momentum map, two pairs of the algebras are naturally isomorphic. We give examples in which all four algebras are not isomorphic.

We consider symmetries of the Dedonder equation arising from variational problems with partial derivatives. Assuming a proper action of the symmetry group, we identify a set of reduced equations on an open dense subset of the domain of definition of the fields under consideration. By continuity, the Dedonder equation is satisfied whenever the reduced equations are satisfied.

Orbits of families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows for a global description of a smooth geometric structure on a family of manifolds in terms of a single object defined on the corresponding family of vector fields. Stratified spaces, Poisson spaces and almost complex spaces are discussed., Comment: 34 pages

This paper presents the theory of Bohr-Sommerfeld-Heisenberg quantization of a completely integrable Hamiltonian system in the context of geometric quantization. The theory is illustrated with several examples.

The not-quite-Hamiltonian theory of singular reduction and reconstruction is described. This includes the notions of both regular and collective Hamiltonian reduction and reconstruction., Comment: 12 pages

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.

Finite time existence and uniqueness of solutions of the evolution equations of minimally coupled Yang-Mills and Dirac systems are proved for inhomogeneous boundary conditions. A characterization of the space of solutions of minimally coupled Yang-Mills and Dirac equations is obtained in terms of the boundary data and the Cauchy data satisfying the constraint equation. The proof is based on a special gauge fixing and a singular perturbation result for the existence of continuous semigroups.

Constructions introduced by Dirac for singular Lagrangians are extended and reinterpreted to cover cases when kernel distributions are either nonintegrable or of nonconstant rank, and constraint sets need not be closed., 30 pages

In this book the author illustrates the power of the theory of subcartesian differential spaces for investigating spaces with singularities. Part I gives a detailed and comprehensive presentation of the theory of differential spaces, including integration of distributions on subcartesian spaces and the structure of stratified spaces. Part II presents an effective approach to the reduction of symmetries. Concrete applications covered in the text include reduction of symmetries of Hamiltonian systems, non-holonomically constrained systems, Dirac structures, and the commutation of quantization with reduction for a proper action of the symmetry group. With each application the author provides an introduction to the field in which relevant problems occur. This book will appeal to researchers and graduate students in mathematics and engineering.

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.

This paper presents a Hamiltonian treatment of nonholonomically constrained mechanical systems. We assume that the total energy of the systems is a sum of kinetic and potential energies and that the constraints are linear in velocities. We prove a nonholonomic version of Noether's theorem and treat a special case of the nonholonomic reconstruction problem. The entire theory is illustrated by a disc, which rolls on a plane without slipping.

We ask whether Hamiltonian vector fields defined on spaces of constant Gaussian curvature $\kappa$ (spheres, for $\kappa>0$, and hyperbolic spheres, for $\kappa, Comment: 12 pages

The possibility of constructing gauge invariant charge densities in classical gauge theory is investigated. For the Yang-Mills-Dirac fields in the Minkowski space, it is shown that, unless the colour label belongs to the centre of the colour algebra, gauge invariant charge densities depend non-locally on the field variables and have non-local Poisson brackets. A quantization leading to observable charge density operators with commutators related to the classical Poisson bracket would violate the locality axiom of quantum field theory. For the Yang-Mills-Dirac fields in a bag the colour algebra is trivial and the colour charges vanish identically. In this sense the Yang-Mills-Dirac theory has to be considered as confining on the basis of a purely mathematical reasoning.

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.

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.