Ewa Eliza Rożko

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.

Dynamics of affinely-rigid bodies with degenerate dimension

Dynamical models with degrees of freedom ruled by linear and affine groups have been discussed in various aspects by many people, e.g. O. I. Bogoyavlensky [7] and J. J. Sŀawianowski [4, 5]. We concentrate on models with degenerate dimension, when the configuration space consists of injections from ℝ m into ℝ n . Equations of motion are derived with special stress on the stationary solutions (stationary ellipses). The isotropic models are related in an interesting way to the theory of Grassmann and stiefel manifolds.

Quantization of affinely-rigid bodies with degenerate dimension

Discussed is Schrodinger quantization of the classical object the configuration space of which is given by the manifold of affine injections from the “material space” into the “physical space” of higher dimension. The special stress is laid on the physical 2- and 3-dimensional situation and on the dynamical models isotropic both in space and material. Basing on the Peter-Weyl theorem we show how to reduce effectively the problem from 6 to 2 degrees of freedom, where as yet no further reduction or separation is found.

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

Quantization of Systems with Internal Degrees of Freedom in Two-Dimensional Manifolds

Presented is a primary step towards quantization of infinitesimal rigid body moving in a two-dimensional manifold. The special stress is laid on spaces of constant curvature like the two-dimensional sphere and pseudosphere (Lobachevsky space). Also two-dimensional torus is briefly discussed as an interesting algebraic manifold.

Quantized mechanics of affinely rigid bodies

In this paper, we develop the main ideas of the quantized version of affinely rigid (homogeneously deformable) motion. We base our consideration on the usual Schrodinger formulation of quantum mechanics in the configuration manifold, which is given, in our case, by the affine group or equivalently by the semi-direct product of the linear group GL(n,R) and the space of translations Rn, where n equals the dimension of the “physical space.” In particular, we discuss the problem of dynamical invariance of the kinetic energy under the action of the whole affine group, not only under the isometry subgroup. Technically, the treatment is based on the 2-polar decomposition of the matrix of the internal configuration and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of elastic media and microstructured continua.

Affinely Rigid Body and Affine Invariance in Physics

Discussed is the dynamics of affinely rigid body, i.e. of a mechanical system the configuration space of which is, roughly speaking, identical with the affine group. So, it is a system placed between two kinds of Euler equations: the rigid body and the ideal incompressible fluids. An essential novelty is our stress on models with the affinely-invariant kinetic energy. It turns out, it may be a toy model towards discussing the problem of affine, non-metrical invariance in fundamental physics, quantum theory and gravitation, and even the nuclear and cosmic physics. Quite independently of that, it may be shown that the affine geodetic model is able to describe the bounded vibrations of classical continua without any help of the external potential energy.

Some Constraints and Symmetries in Dynamics of Homogeneously Deformable Elastic Bodies

Our work has been inspired among others by the work of Arnold, Kozlov and Neihstadt. Our goal is to carry out a thorough analysis of the geometric problems we are faced with in the dynamics of affinely rigid bodies. We examine two models: classical dynamics description by d’Alembert and vakonomic one. We conclude that their results are quite different. It is not yet clear which model is practically better.

Quasiclassical and Quantum Dynamics of Systems of Angular Momenta

Here we use the mathematical structure of group algebras and H-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are SU(2) and its quotient SO(3,R). The proposed scheme is considered in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital and spin angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules and etc. Secondly, the other promising area of applications is Schrodinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schrodinger framework the algebras of operators related to group algebras are a very useful tool. Finally, we investigate also some problems of composed systems and the quasiclassical limit obtained as the asymptotics of “large” quantum numbers, i.e., “quickly oscillating” wave functions on groups. They are related in an interesting way to the geometry of the coadjoint orbits of the Lie group SU(2). The presentation is based on the general ideas of applying group-algebraic methods and extesive use of the Lie group structure. The papers ends with consideration of the special case of the group SU(2) and its quotient SO(3,R), which is the main subject in this paper, i.e., angular momentum problems. Formally, the scheme could be applied to the isospin systems. However, it is rather hard to imagine realistic quasiclassical isospin problems.