Barbara Gołubowska

Affine symmetry in mechanics of collective and internal modes. Part II. Quantum models

Abstract Discussed is the quantized version of the classical description of collective and internal affine modes as developed in Part I. We perform the Schrodinger quantization and reduce effectively the quantized problem from n 2 to n degrees of freedom. Some possible applications in nuclear physics and other quantum many-body problems are suggested. Discussed is also the possibility of half-integer angular momentum in composed systems of spinless particles.

Affine symmetry in mechanics of collective and internal modes. Part I. Classical models

Discussed is a model of collective and internal degrees of freedom with kinematics based on affine group and its subgroups. The main novelty in comparison with the previous attempts of this kind is that it is not only kinematics but also dynamics that is affinely-invariant. The relationship with the dynamics of integrable one-dimensional lattices is discussed. It is shown that affinely-invariant geodetic models may encode the dynamics of something like elastic vibrations.

Affine models of internal degrees of freedom and their action-angle description

Abstract We discuss equations of motion of small affinely-rigid bodies in the action-angle description. By “small” we mean a body of infinitesimal size moving in a Riemann space. Roughly speaking the extended body is replaced by the structured material point with internal degrees of freedom. We discuss some special cases of constant curvature two-dimensional spaces, like sphere and pseudosphere. Besides, motion on the torus manifold is discussed.

Models of internal degrees of freedom based on classical groups and their homogeneous spaces

Abstract We derive and discuss equations of motion of an infinitesimal affinely-rigid body moving in a Riemannian space. The main novelty here is that the internal motion is described with respect to some prescribed field of orthonormal frames. This enables us to describe effectively interactions between internal rotations and homogeneous deformations. One can obtain smoothly the limiting rigid body case and discuss the problem of small elastic deformations imposed on rigid rotations. Some physically realistic and integrable models in two dimensions are discussed. We were motivated by the pure analytical mechanics and theory of dynamical systems, nevertheless some physical applications are not excluded, e.g. the motion of continental plates on the surface of the Earth.

Motion of test rigid bodies in Riemannian spaces

Abstract We derive and discuss equations of motion of rigid bodies of infinitesimal size in a Riemann space. The treatment is nonrelativistic. The rigorous meaning of “infinitisemal size” consists in replacing an extended body by the structured material point with internal degrees of freedom (co-moving orthonormal frame). The special case of constant curvature two-dimensional spaces is discussed and that of three-dimensional ones mentioned. The problem reduces then to Hamiltonian systems with the higher-dimensional rotation and Lorentz groups as configuration spaces.

Action-Angle Analysis of Some Geometric Models of Internal Degrees of Freedom

Abstract We derive and discuss equations of motion of infinitesimal affinely-rigid body moving in Riemannian spaces. There is no concept of extended rigid and affinely rigid body in a general Riemannian space. Therefore the gyroscopes with affine degrees of freedom are described as moving bases attached to the material point. This base is a remnant of extended rigid and affinely rigid body in a flat space. The special stress is laid on affinely rigid bodies in two-dimensional constant curvature spaces (sphere and pseudosphere). In particular, we consider incompressible affinely rigid bodies, like, e.g. fat spots on a water surface (e.g. petrol pollution). This is a two-dimensional analogue of three-dimensional incompressible objects like fluid droplets.

Mechanics of systems of affine bodies. Geometric foundations and applications in dynamics of structured media

Discussed are geometric structures underlying analytical mechanics of systems of affine bodies. Presented is detailed algebraic and geometric analysis of concepts like mutual deformation tensors and their invariants. Problems of affine invariance and of its interplay with the usual Euclidean invariance are reviewed. This analysis was motivated by mechanics of affine (homogeneously deformable) bodies, nevertheless, it is also relevant for the theory of unconstrained continua and discrete media. Postulated are some models where the dynamics of elastic vibrations is encoded not only in potential energy (sometimes even not at all) but also (sometimes first of all) in appropriately chosen models of kinetic energy (metric tensor on the configuration space), like in Maupertuis principle. Physically, the models may be applied in structured discrete media, molecular crystals, fullerens, and even in description of astrophysical objects. Continuous limit of our affine-multibody theory is expected to provide a new class of micromorphic media. Copyright

Motion of test bodies with internal degrees of freedom in non-Euclidean spaces

Discussed is mechanics of objects with internal degrees of freedom in generally non-Euclidean spaces. Geometric peculiarities of the model are investigated in detail Discussed are also possible mechanical applications, e.g. in dynamics of structured continua, defect theory and in other fields of mechanics of deformable bodies. Elaborated is a new method of analysis based on nonholonomic frames. We compare our results and methods with those of other authors working in nonlinear dynamics. Simple examples are presented.

Constraints and symmetry in mechanics of affine motion

Abstract The aim of this paper is to perform a deeper geometric analysis of problems appearing in dynamics of affinely rigid bodies. First of all we present a geometric interpretation of the polar and the two-polar decomposition of affine motion. Later on some additional constraints imposed on the affine motion are reviewed, both holonomic and non-holonomic. In particular, we concentrate on certain natural non-holonomic models of the rotation-less motion. We discuss both the usual d’Alembert model and the vakonomic dynamics. The resulting equations are quite different. It is not yet clear which model is practically better. In any case they both are different from the holonomic constraints defining the rotation-less motion as a time-dependent family of symmetric matrices of placements. The latter model seems to be non-geometric and non-physical. Nevertheless, there are certain relationships between our non-holonomic models and the polar decomposition.

Generalized Weyl–Wigner–Moyal–Ville Formalism and Topological Groups

Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal, and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on, we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of ‘Umklapp–Prozessen’. Copyright