40 research outputs found
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
Mechanics of Systems of Affine Bodies. Geometric Foundations and Applications in Dynamics of Structured Media
In the present paper we investigate the mechanics of systems of
affinely-rigid bodies, i.e., bodies rigid in the sense of affine geometry.
Certain physical applications are possible in modelling of molecular crystals,
granular media, and other physical objects. Particularly interesting are
dynamical models invariant under the group underlying geometry of degrees of
freedom. In contrary to the single body case there exist nontrivial potentials
invariant under this group (left and right acting). The concept of relative
(mutual) deformation tensors of pairs of affine bodies is discussed. Scalar
invariants built of such tensors are constructed. There is an essential novelty
in comparison to deformation scalars of single affine bodies, i.e., there exist
affinely-invariant scalars of mutual deformations. Hence, the hierarchy of
interaction models according to their invariance group, from Euclidean to
affine ones, can be considered.Comment: 50 pages, 4 figure
Geometric Nonlinearities in Field Theory, Condensed Matter and Analytical Mechanics
There are two very important subjects in physics: Symmetry of dynamical
models and nonlinearity. All really fundamental models are invariant under some
particular symmetry groups. There is also no true physics, no our Universe and
life at all, without nonlinearity. Particularly interesting are essential,
non-perturbative nonlinearities which are not described by correction terms
imposed on some well-defined linear background. Our idea in this paper is that
there exists some mysterious, not yet understood link between essential,
physically relevant nonlinearity and dynamical symmetry, first of all, large
symmetry groups. In some sense the problem is known even in soliton theory,
where the essential nonlinearity is often accompanied by the infinite system of
integrals of motion, thus, by infinite-dimensional symmetry groups. Here we
discuss some more familiar problems from the realm of field theory, condensed
matter physics, and analytical mechanics, where the link between essential
nonlinearity and high symmetry is obvious, even if not yet fully understood.Comment: 26 page
Affine symmetry in mechanics of collective and internal modes. Part II. Quantum models
Discussed is the quantized version of the classical description of collective
and internal affine modes as developed in Part I. We perform the Schr\"odinger
quantization and reduce effectively the quantized problem from to
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 spin-less particles