Jȩdrzej Śniatycki

Geometry of nonholonomic constraints

Abstract 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 Noethers 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.

Nonholonomic Noether theorem and reduction of symmetries

Abstract A necessary and sufficient condition for a function to be a constant of motion of a Lagrangian system with nonlinear nonholonomic constraints is derived in the framework of the Hamiltonian formulation. It is shown that the symmetries generated by constants of motion form a normal subgroup G 0 of the symmetry group G of the theory. The reduction of symmetries, taking into account that G 0 is generated by constants of motion, decreases the number of the dynamical variables by dim G + dim G μ , where G μ ⊂ G 0 is the isotropy group of the value μ of the conserved momentum.

Local integrability of the mixmaster model

Abstract In this paper we study the mixmaster universe whose evolution is described by a polynomial Hamiltonian. We show that this model is locally integrable. The Taub solutions form an integrable subsystem. We show that there are no solutions of the mixmaster universe asymptotic to a partially isotropic gravitational collapse, that is, a collapse where two of the metric coefficients are always equal, other than the Taub solutions.

Almost poisson spaces and nonholonomic singular reduction

Abstract Dynamics of Hamiltonian systems with linear nonholonomic constraints is described by distributional Hamiltonian systems. We show that the space of orbits of a proper action of a symmetry group of a distributional Hamiltonian system is a differential space partitioned by smooth manifolds preserved by the evolution. The reduced dynamics is given by distributional Hamiltonian systems on the projections of the manifolds of the partition. It is described in terms of the almost Poisson algebra of smooth functions on the orbit space.

The cauchy data space formulation of classical field theory

Abstract The space C of Cauchy data essentially consists of all Cauchy data on all Cauchy surfaces for the field theory under consideration. There is a canonically defined distribution on C , the integral manifolds of which represent the histories of the field. This formulation of field theory is purely intrinsic in contradistinction to the standard Hamiltonian formulation which requires a 3 + 1 splitting of the space time.

Bohr-Sommerfeld conditions in geometric quantization

Abstract The corrected Bohr-Sommerfeld quantum conditions, ∫ p dq−d = integer, are studied in the framework of geometric quantization. It is shown, in the representation given by a polarization F, that a half-form corresponds to a wave function only if it vanishes on all closed curves with tangent vectors in F for which the quantum condition is not satisfied. The constant d is determined, for each closed curve y, by the element of the holonomy group of a bundle of metalinear frames for F induced by y. This result is applied to a one-dimensional harmonic oscillator and a two-dimensional relativistic Kepler problem. In the case of the one-dimensional harmonic oscillator there are two possibilities of choosing a metalinear frame bundle for F. One choice leads to the original Bohr-Sommerfeld condition while the other leads to the corrected version with d = 1 2 . Similarly, choosing different metalinear frame bundles for F, we get for the relativistic Kepler problem the fine structures of the energy levels corresponding to spin 0 and spin 1 2 .

The momentum equation and the second order differential equation condition

Abstract It is shown that equations of motion of mechanical systems with nonholonomic constraints are equivalent to the momentum equation for all vector fields on the configuration space with values in the constraint distribution, and the second-order differential equation condition for all smooth functions on the configuration space. For a free and proper action of the symmetry group on the configuration space, the reduced equations of motion are equivalent to the momentum equation for all invariant vector fields on the configuration space with values in the constraint distribution and the second-order differential equation condition for all invariant functions on the configuration space. Applications to singular reduction are discussed.

How to get masses from Kaluza-Klein theory

Abstract In the general Kaluza-Klein theory on a principal bundle P with structure group G, the vertical part of the metric g on P is defined by a scalar field γ on P. We consider γ as a Higgs field but, instead of looking for an appropriate potential, we constrain this scalar field to a G -orbit. This procedure provides symmetry breaking and the Yang-Mills part of the fields split into a residual Yang-Mills field and a set of vector bosons which are shown to be massive. We write down the field equations for gravity, Yang-Mills and massive gauge bosons. An example with internal symmetry group Spin (4) broken to U (1) is worked out.

QUANTIZATION OF SINGULAR REDUCTION

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 Kahler 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.

A geometric interpretation of symmetry breaking in electroweak interactions

Abstract Symmetry breaking mechanism, in which the Higgs potential is replaced by a constraint restricting the Higgs field to an orbit, is discussed in the framework of the standard model. On the classical level, this approach gives the same physical predictions as the standard model, except that it does not require the existence of a Higgs boson.