dr hab. Wasyl Kowalczuk

Zakład Teorii Ośrodków Ciągłych i Nanostruktur (ZTOCiN)
Pracownia Mechaniki Analitycznej i Teorii Pola (PMAiTP)
stanowisko: adiunkt
telefon: (+48) 22 826 12 81 wew.: 135
pokój: 114
e-mail: vkoval

Doktorat
2006-03-31Nonlinear models of collective and internal degrees of freedom in mechanics and field theory 
promotor -- prof. dr hab. Jan Sławianowski, IPPT PAN
591
 
Habilitacja
2015-06-25Modele afiniczne w opisie dyskretnych i ciągłych ośrodków z mikrostrukturą w mechanice analitycznej 
Ostatnie publikacje
1.Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Mechanics of affine bodies. Towards affine dynamical symmetry, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, ISSN: 0022-247X, DOI: 10.1016/j.jmaa.2016.08.042, Vol.446, pp.493-520, 2017

Streszczenie:

In this paper we discuss certain dynamical models of affine bodies, including problems of partial separability and integrability. There are some reasons to expect that the suggested models are dynamically viable and that on the fundamental level of physical phenomena the “large” affine symmetry of dynamical laws is more justified and desirable than the restricted invariance under isometries.

Słowa kluczowe:

Homogeneous deformation, structured media, affinely-invariant dynamics, elastic vibrations encoded in kinetic energy, Calogero-Moser and Sutherland integrable lattices

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
35p.
2.Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Quantized mechanics of affinely-rigid bodies, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.4501, pp.1-19, 2017

Streszczenie:

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 Schrödinger formulation of quantum mechanics in the configurationmanifold, 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 R^n, 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.

Słowa kluczowe:

Homogeneously deformable body, Peter-Weyl analysis, Schrödinger quantization.

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
25p.
3.Gołubowska B., Kovalchuk V., Sławianowski J.J., Constraints and symmetry in mechanics of affine motion, JOURNAL OF GEOMETRY AND PHYSICS, ISSN: 0393-0440, DOI: 10.1016/j.geomphys.2014.01.012, Vol.78, pp.59-79, 2014

Streszczenie:

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.

Słowa kluczowe:

Affine motion, polar and two-polar decompositions, non-holonomic constraints, d’Alembert and Lusternik variational principles, vakonomic constraints

Afiliacje autorów:

Gołubowska B.-IPPT PAN
Kovalchuk V.-IPPT PAN
Sławianowski J.J.-IPPT PAN
25p.
4.Popov A., Kovalchuk V., Parametric representation of wave propagation in non-uniform media (both in transmission and stop bands), MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.2687, Vol.36, No.11, pp.1350-1362, 2013

Streszczenie:

An analytical approach based on the parametric representation of the wave propagation in non-uniform media was considered. In addition to the previously developed theory of parametric antiresonance describing the field attenuation in stop bands, in the present paper, the behaviour of the Bloch wave in a transmission band was investigated. A wide class of exact solutions was found, and the correspondence to the quasi-periodic Floquet solutions was shown.

Słowa kluczowe:

Floquet theorem, parametric resonance/antiresonance, wave propagation, non-uniform media, periodic structures, stop and transmission bands

Afiliacje autorów:

Popov A.-Pushkov Institute of Terrestrial Magnetism, Ionosphere and RadioWave Propagation, Russian Academy of Sciences (RU)
Kovalchuk V.-IPPT PAN
25p.
5.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, ISSN: 1531-3492, DOI: 10.3934/dcdsb.2012.17.699, Vol.17, No.2, pp.699-733, 2012

Streszczenie:

The main leitmotivs of this paper are the essential nonlinearities, symmetries and the mutual relationships between them. By essential nonlinearities we mean ones which are not interpretable as some extra perturbations imposed on a linear background deciding about the most important qualitative features of discussed phenomena. We also investigate some discrete and continuous systems, roughly speaking with large symmetry groups. And some remarks about the link between two concepts are reviewed. Namely, we advocate the thesis that the most important non-perturbative nonlinearities are those implied by the assumed “large” symmetry groups. It is clear that such a relationship does exist, although there is no complete theory. We compare the mechanism of inducing nonlinearity by symmetry groups of discrete and continuous systems. Many striking and instructive analogies are found, e.g., analogy between analytical mechanics of systems of affine bodies and general relativity, tetrad models of gravitation, and Born-Infeld nonlinearity. Some interesting, a bit surprising problems concerning Noether theorem are discussed, in particular in the context of large symmetry groups.

Słowa kluczowe:

Essential nonlinearity, dynamical symmetries, affine bodies, general relativity, tetrad models and micromorphic continua

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
30p.
6.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Generalized Weyl–Wigner–Moyal–Ville formalism and topological groups, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.1531, Vol.35, pp.17-42, 2012

Streszczenie:

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’.

Słowa kluczowe:

Weyl–Wigner–Moyal–Ville formalism, topological groups, classical momentum mapping, unitary projective representations, Pontryagin duality, ‘Umklapp–Prozessen’

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
25p.
7.Gołubowska B., Kovalchuk V., Martens A., Rożko E.E., Sławianowski J.J., Some strange features of the Galilei group, Journal of Geometry and Symmetry in Physics, ISSN: 1312-5192, DOI: 10.7546/jgsp-26-2012-33-59, Vol.26, pp.33-59, 2012

Streszczenie:

Discussed are certain strange properties of the Galilei group, connected first of all with the property of mechanical energy-momentum covector to be an affine object, rather than the linear one. Its affine transformation rule is interesting in itself and dependent on the particle mass. On the quantum level this means obviously that we deal with the projective unitary representation of the group rather than with the usual representation. The status of mass is completely different than in relativistic theory, where it is a continuous eigenvalue of the Casimir invariant. In Galilei framework it is a parameter characterizing the factor of the projective representation, in the sense of V. Bargmann. This “pathology” from the relativistic point of view is nevertheless very interesting and it underlies the Weyl-Wigner-Moyal-Ville approach to quantum mechanics.

Słowa kluczowe:

Galilei group, affine transformation, particle mass, projective unitary representation, Weyl-Wigner-Moyal-Ville formalism

Afiliacje autorów:

Gołubowska B.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
Sławianowski J.J.-IPPT PAN
8.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Mechanics of systems of affine bodies. Geometric foundations and applications in dynamics of structured media, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.1462, Vol.34, pp.1512-1540, 2011

Streszczenie:

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.

Słowa kluczowe:

Systems of affine bodies, mutual deformation tensors, affine and Euclidean invariance, structured media, elastic vibrations, geometric structures

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
25p.
9.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Quasiclassical and Quantum Systems of Angular Momentum. Part I. Group Algebras as a Framework for Quantum-Mechanical Models with Symmetries, Journal of Geometry and Symmetry in Physics, DOI: 10.7546/jgsp-21-2011-61-94, Vol.21, pp.61-94, 2011

Streszczenie:

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 applied 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. Secondly, the other promising area of applications is Schroedinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schroedinger framework the algebras of operators related to group algebras are a very useful tool. We investigate 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 geometry of the coadjoint orbits of SU(2).

Słowa kluczowe:

Systems of angular momenta, models with symmetries, quantum dynamics, spin systems, Schroedinger quantum mechanics, quasiclassical limit

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
10.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Quasiclassical and Quantum Systems of Angular Momentum. Part II. Quantum Mechanics on Lie Groups and Methods of Group Algebras, Journal of Geometry and Symmetry in Physics, DOI: 10.7546/jgsp-22-2011-67-94, Vol.22, pp.67-94, 2011

Streszczenie:

In Part I of this series we have presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment there was very “ascetic” in that only the structure of a locally compact topological group was used. Below we explicitly make use of the Lie group structure. Relying on differential geometry one is able to introduce explicitly representation of important physical quantities and to formulate the general ideas of quasiclassical representation and classical analogy.

Słowa kluczowe:

Systems of angular momenta, models with symmetries, quantum dynamics, Lie group structures, quasiclassical representation, classical analogy

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
11.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Quasiclassical and Quantum Systems of Angular Momentum. Part III. Group Algebra su(2), Quantum Angular Momentum and Quasiclassical Asymptotics, Journal of Geometry and Symmetry in Physics, DOI: 10.7546/jgsp-23-2011-59-95, Vol.23, pp.59-95, 2011

Streszczenie:

This is the third part of our series “Quasiclassical and Quantum Systems of Angular Momentum”. In two previous parts we have discussed the methods of group algebras in formulation of quantum mechanics and certain quasiclassical problems. Below we specify to the special case of the group SU(2) and its quotient SO(3,R), and discuss just our main subject in this series, i.e., angular momentum problems. To be more precise, this is the purely SU(2)-treatment, so formally this might also apply to isospin. However. it is rather hard to imagine realistic quasiclassical isospin problems.

Słowa kluczowe:

Systems of angular momenta, models with symmetries, quantum dynamics, quasiclassical isospin problems

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
12.Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Quantized excitations of internal affine modes and their influence on Raman spectra, ACTA PHYSICA POLONICA B, ISSN: 0587-4254, DOI: 10.5506/APhysPolB.41.165, Vol.41, No.1, pp.165-218, 2010

Streszczenie:

Discussed is the structure of classical and quantum excitations of internal degrees of freedom of multiparticle objects like molecules, fullerens, atomic nuclei, etc. Basing on some invariance properties under the action of isometric and affine transformations we reviewed some new models of the mutual interaction between rotational and deformative degrees of freedom. Our methodology and some results may be useful in the theory of Raman scattering and nuclear radiation.

Słowa kluczowe:

Internal degrees of freedom, isometric and affine invariance, classical and quantum excitations, multiparticle objects, Raman scattering, nuclear radiation

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
20p.
13.Sławianowski J.J., Kovalchuk V., Schrödinger and related equations as Hamiltonian systems, manifolds of second-order tensors and new ideas of nonlinearity in quantum mechanics, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, DOI: 10.1016/S0034-4877(10)00008-X, Vol.65, No.1, pp.29-76, 2010

Streszczenie:

Considered is the Schrödinger equation in a finite-dimensional space as an equation of mathematical physics derivable from the variational principle and treatable in terms of the Lagrange-Hamilton formalism. It provides an interesting example of “mechanics” with singular Lagrangians, effectively treatable within the framework of Dirac formalism. We discuss also some modified “Schrödinger” equations involving second-order time derivatives and introduce a kind of nondirect, nonperturbative, geometrically-motivated nonlinearity based on making the scalar product a dynamical quantity. There are some reasons to expect that this might be a new way of describing open dynamical systems and explaining some quantum “paradoxes”.

Słowa kluczowe:

Hamiltonian systems on manifolds of scalar products, finite-level quantum systems, finite-dimensional Hilbert space, Hermitian forms, scalar product as a dynamical variable, Schroedinger equation, Dirac formalism, essential nonperturbative nonlinearity, quantum paradoxes, conservation laws, GL_n(C)-invariance

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
13p.
14.Kovalchuk V., On classical dynamics of affinely-rigid bodies subject to the Kirchhoff-Love constraints, Symmetry, Integrability and Geometry: Methods and Applications SIGMA, ISSN: 1815-0659, DOI: 10.3842/SIGMA.2010.031, Vol.6, No.031, pp.1-12, 2010

Streszczenie:

In this article we consider the affinely-rigid body moving in the three-dimensional physical space and subject to the Kirchhoff–Love constraints, i.e., while it deforms homogeneously in the two-dimensional central plane of the body it simultaneously performs one-dimensional oscillations orthogonal to this central plane. For the polar decomposition we obtain the stationary ellipsoids as special solutions of the general, strongly nonlinear equations of motion. It is also shown that these solutions are conceptually different from those obtained earlier for the two-polar (singular value) decomposition.

Słowa kluczowe:

Affinely-rigid bodies with degenerate dimension, Kirchhoff–Love constraints, polar decomposition, Green deformation tensor, deformation invariants, stationary ellipsoids as special solutions

Afiliacje autorów:

Kovalchuk V.-IPPT PAN
15.Kovalchuk V., Rożko E.E., Classical models of affinely-rigid bodies with "thickness" in degenerate dimension, Journal of Geometry and Symmetry in Physics, ISSN: 1312-5192, DOI: 10.7546/jgsp-14-2009-51-65, Vol.14, pp.51-65, 2009

Streszczenie:

The special interest is devoted to such situations when the material space of object with affine degrees of freedom has generally lower dimension than the one of the physical space. In other words when we have m-dimensional affinely-rigid body moving in the n-dimensional physical space, m < n. We mainly concentrate on the physical situation m = 2, n = 3 when “thickness” of flat bodies performs one-dimensional oscillations orthogonal to the two-dimensional central plane of the body. For the isotropic case in two “flat” dimensions some special solutions, namely, the stationary ellipses, which are analogous to the ellipsoidal figures of equilibrium well known in astro- and geophysics, e.g., in the theory of the Earth’s shape, are obtained.

Słowa kluczowe:

Affinely-rigid bodies with degenerate dimension, non-zeroth thickness, two-polar decomposition, stationary ellipsoidal figures of equilibrium

Afiliacje autorów:

Kovalchuk V.-IPPT PAN
Rożko E.E.-IPPT PAN
16.Kovalchuk V., Sławianowski J.J., Hamiltonian systems inspired by the Schrödinger equation, Symmetry, Integrability and Geometry: Methods and Applications SIGMA, ISSN: 1815-0659, DOI: 10.3842/SIGMA.2008.046, Vol.4, pp.46-54, 2008

Streszczenie:

Described is n-level quantum system realized in the n-dimensional “Hilbert” space H with the scalar product G taken as a dynamical variable. The most general Lagrangian for the wave function and G is considered. Equations of motion and conservation laws are obtained. Special cases for the free evolution of the wave function with fixed G and the pure dynamics of G are calculated. The usual, first- and second-order modified Schroedinger equations are obtained.

Słowa kluczowe:

Schroedinger equation, Hamiltonian systems on manifolds of scalar products, n-level quantum systems, scalar product as a dynamical variable, essential non-perturbative nonlinearity, conservation laws, GL_n(C)-invariance

Afiliacje autorów:

Kovalchuk V.-IPPT PAN
Sławianowski J.J.-IPPT PAN
17.Sławianowski J.J., Kovalchuk V., Sławianowska A., Gołubowska B., Martens A., Rożko E.E., Zawistowski Z.J., Affine symmetry in mechanics of collective and internal modes. Part II. Quantum models, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, DOI: 10.1016/S0034-4877(05)80002-3, Vol.55, No.1, pp.1-46, 2005

Streszczenie:

Discussed is the quantized version of the classical description of collective and internal affine modes as developed in Part I. We perform the Schroedinger 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.

Słowa kluczowe:

Collective modes, affine invariance, Schroedinger quantization, quantum many-body problem

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Sławianowska A.-other affiliation
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
Zawistowski Z.J.-IPPT PAN
18.Sławianowski J.J., Kovalchuk V., Sławianowska A., Gołubowska B., Martens A., Rożko E.E., Zawistowski Z.J., Affine symmetry in mechanics of collective and internal modes. Part I. Classical models, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, DOI: 10.1016/S0034-4877(04)80026-0, Vol.54, No.3, pp.373-427, 2004

Streszczenie:

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.

Słowa kluczowe:

Collective modes, affine invariance, integrable lattices, nonlinear elasticity

Afiliacje autorów:

Sławianowski J.J.-other affiliation
Kovalchuk V.-IPPT PAN
Sławianowska A.-other affiliation
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
Zawistowski Z.J.-IPPT PAN
19.Sławianowski J.J., Kovalchuk V., Invariant geodetic problems on the projective group Pr(n,R), Proceedings of Institute of Mathematics of NAS of Ukraine, Vol.50, No.2, pp.955-960, 2004

Streszczenie:

The concept of n-dimensional projectively-rigid body is introduced and its connection to the concept of (n+ 1)-dimensional incompressible affinely-rigid body is analysed. The equations of geodetic motion for such a projectively-rigid body are obtained. As an instructive example, the special case of n = 1 is investigated.

Słowa kluczowe:

Projectively-rigid bodies, incompressible affinely-rigid bodies, left and right invariant problems, geodetic motion

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
20.Kovalchuk V., Green function for Klein-Gordon-Dirac equation, Journal of Nonlinear Mathematical Physics, ISSN: 1402-9251, DOI: 10.2991/jnmp.2004.11.s1.9, Vol.11, No.Supplement, pp.72-77, 2004

Streszczenie:

The Green function for Klein-Gordon-Dirac equation is obtained. The case with the dominating Klein-Gordon term is considered. There seems to be a formal analogy between our problem and a certain problem for a 4-dimensional particle moving in the external field. The explicit relations between the wave function, Green function and initial conditions are established with the help of the T-exponent formalism.

Słowa kluczowe:

Green functions, Klein-Gordon-Dirac equation, T-exponent formalism, Dirac plane harmonic waves

Afiliacje autorów:

Kovalchuk V.-IPPT PAN
21.Sławianowski J.J., Kovalchuk V., Classical and quantized affine physics: a step towards it, Journal of Nonlinear Mathematical Physics, ISSN: 1402-9251, DOI: 10.2991/jnmp.2004.11.s1.21, Vol.11, No.Supplement, pp.157-166, 2004

Streszczenie:

The classical and quantum mechanics of systems on Lie groups and their homogeneous spaces are described. The special stress is laid on the dynamics of deformable bodies and the mutual coupling between rotations and deformations. Deformative modes are discretized, i.e., it is assumed that the relevant degrees of freedom are controlled by a finite number of parameters. We concentrate on the situation when the effective configuration space is identical with affine group (affinely-rigid bodies). The special attention is paid to left- and right-invariant geodetic systems, when there is no potential term and the metric tensor underlying the kinetic energy form is invariant under left or/and right regular translations on the group. The dynamics of elastic vibrations may be encoded in this way in the very form of kinetic energy. Although special attention is paid to invariant geodetic systems, the potential case is also taken into account.

Słowa kluczowe:

Lie groups, homogeneous spaces, defomable bodies, left and right affine invariance, geodetic models, classical and quantum mechanics

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
22.Sławianowski J.J., Kovalchuk V., Invariant geodetic problems on the affine group and related Hamiltonian systems, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, DOI: 10.1016/S0034-4877(03)80029-0, Vol.51, No.2/3, pp.371-379, 2003

Streszczenie:

Discussed are (pseudo-)Riemannian metrics on the affine group. A special stress is laid on metric structures invariant under left or right regular translations by elements of the total affine group or some of its geometrically distinguished subgroups. Also some non-geodetic problems in corresponding Riemannian spaces are discussed.

Słowa kluczowe:

Affinely-rigid body, Riemannian metrics, geodetic problems, two-polar decomposition, Hamiltonian systems

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
23.Sławianowski J.J., Kovalchuk V., Klein-Gordon-Dirac equation: physical justification and quantization attempts, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, DOI: 10.1016/S0034-4877(02)80023-4, Vol.49, No.2/3, pp.249-257, 2002

Streszczenie:

Discussed is the Klein-Gordon-Dirac equation, i.e. a linear differential equation with constant coefficients, obtained by superposing Dirac and d'Alembert operators. A general solution of KGD equation as a superposition of two Dirac plane harmonic waves with different masses has been obtained. The multiplication rules for Dirac bispinors with different masses have been found. Lagrange formalism has been applied to receive the energy-momentum tensor and 4-current. It appears, in particular, that the scalar product is a superposition of Klein-Gordon and Dirac scalar products. The primary approach to canonical formalism is suggested. The limit cases of equal masses and one zero mass have been calculated.

Słowa kluczowe:

Klein-Gordon-Dirac equation, plane harmonic waves with different masses, Dirac bispinors, Lagrange formalism, canonical formalism

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN

Lista ostatnich monografii
1.
67
Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Dynamical systems with internal degrees of freedom in non-Euclidean spaces, IPPT Reports on Fundamental Technological Research, 8, pp.1-129, 2006
2.
484
Sławianowski J.J., Kovalchuk V., Sławianowska A., Gołubowska B., Martens A., Rożko E.E., Zawistowski Z.J., Invariant geodetic systems on Lie groups and affine models of internal and collective degrees of freedom, Prace IPPT - IFTR Reports, Warszawa, 7, pp.1-164, 2004
Lista rozdziałów w ostatnich monografiach
1.
374
Sławianowski J.J., Kovalchuk V., Selected Topics in Applications of Quantum Mechanics, rozdział: Classical or Quantum? What is Reality?, prof. Mohammad Reza Pahlavani (Ed.), InTech, Rijeka, pp.3-35, 2015
2.
387
Sławianowski J.J., Kovalchuk V., Geometry, Integrability, Mechanics and Quantization, rozdział: Quantized version of the theory of affine body, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.68-88, 2015
3.
391
Kovalchuk V., Geometry, Integrability, Mechanics and Quantization, rozdział: On new ideas of nonlinearity in quantum mechanics, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.267-279, 2015
4.
313
Sławianowski J.J., Kovalchuk V., Advances in Quantum Mechanics, rozdział: Schroedinger equation as a hamiltonian system, essential nonlinearity, dynamical scalar product and some ideas of decoherence, Prof. Paul Bracken (Ed.), InTech, Rijeka, pp.81-103, 2013
5.
252
Gołubowska B., Kovalchuk V., Martens A., Rożko E.E., Sławianowski J.J., Geometry, Integrability and Quantization XIII, rozdział: Some strange features of the Galilei group, Editors: Ivailo M. Mladenov, Andrei Ludu and Akira Yoshioka, Avangard Prima, Sofia, pp.150-175, 2012
6.
35
Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Geometry, Integrability and Quantization XII, rozdział: Quasiclassical and Quantum Dynamics of Systems of Angular Momenta, Editors: Ivailo M. Mladenov, Gaetano Vilasi and Akira Yoshioka, Avangard Prima, Sofia, pp.70-155, 2011
7.
147
Kovalchuk V., Rożko E.E., Geometry, integrability and quantization, Proceedings of the 10th International Conference, rozdział: Classical models of affinely-rigid bodies with thickness in degenerate dimension, Bulgarian Academy of Sciences, Avangard Prima, Mladenov I.M., Vilasi G., Yoshioka A. (Eds.), pp.197-210, 2009
8.
181
Sławianowski J.J., Kovalchuk V., Problems of stability and stabilization of motion, Bulletin of Computing Centre named after A.A.Dorodnitsyn, rozdział: Symmetries and geometrically implied nonlinearities in mechanics and field theory, Russian Academy of Sciences, Stepanov S.Ja., Burov A.A. (Eds.), pp.119-150, 2009
9.
149
Sławianowski J.J., Kovalchuk V., Geometry, integrability and quantization, Proceedings of the 9th International Conference, rozdział: Search for the geometrodynamical gauge group. Hypotheses and some results, Bulgarian Academy of Sciences, Mladenov M.I., De Leon M. (Eds.), pp.63-132, 2008
Redaktor monografii
1.
385
Kovalchuk V., Geometry, Integrability, Mechanics and Quantization, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.1-461, 2015

Prace konferencyjne
1.Sławianowski J.J., Kovalchuk V., Quantized version of the theory of affine body, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-73-93, Vol.XVI, pp.73-93, 2015

Streszczenie:

In the previous lecture we have introduce and discussed the concept of affinely-rigid, i.e., homogeneously deformable body. Some symmetry problems and possible applications were discussed. We referred also to our motivation by Euler ideas. Below we describe the general principles of the quantization of this theory in the Schroedinger language. The special stress is laid on highly-symmetric, in particular affinely-invariant, models and the Peter-Weyl analysis of wave functions.

Słowa kluczowe:

Homogeneously deformable body, Schroedinger quantization, affine invariance, highly symmetric models, Peter-Weyl analysis

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
2.Kovalchuk V., On new ideas of nonlinearity in quantum mechanics, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-195-206, Vol.XVI, pp.195-206, 2015

Streszczenie:

Our main idea is to suggest a new model of non-perturbative and geometrically motivated nonlinearity in quantum mechanics. The Schroedinger equation and corresponding relativistic linear wave equations derivable from variational principles are analyzed as usual self-adjoint equations of mathematical physics. It turns out that introducing the second-order time derivatives to dynamical equations, even as small corrections, can help to obtain the regular Legendre transformation. Following the conceptual transition from the special to general theory of relativity, where the metric tensor loses its status of the absolute geometric object and becomes included into degrees of freedom (gravitational field), in our treatment the Hilbert-space scalar product becomes a dynamical quantity which satisfies together with the state vector the system of differential equations. The structure of obtained Lagrangian and equations of motion is very beautiful, as usually in high-symmetry problems.

Słowa kluczowe:

Non-perturbative nonlinearity, self-adjoint dynamical equations, Schroedinger equation, highly symmetric problems, Hilbert-space scalar product

Afiliacje autorów:

Kovalchuk V.-IPPT PAN
3.Gołubowska B., Kovalchuk V., Rożko E.E., Sławianowski J.J., Some constraints and symmetries in dynamics of homogeneously deformable elastic bodies, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/gip-14-2013-103-115, Vol.XIV, pp.103-115, 2013

Streszczenie:

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 i practically better.

Słowa kluczowe:

Affine motion, non-holonomic constraints, d’Alembert and Lusternik variational principles, vakonomic constraints

Afiliacje autorów:

Gołubowska B.-IPPT PAN
Kovalchuk V.-IPPT PAN
Rożko E.E.-IPPT PAN
Sławianowski J.J.-IPPT PAN