Wasyl Kowalczuk, Ph.D., Dr. Habil. 

Doctoral thesis
20060331  Nonlinear models of collective and internal degrees of freedom in mechanics and field theory
 591 
Habilitation thesis
20150625  Modele afiniczne w opisie dyskretnych i ciągłych ośrodków z mikrostrukturą w mechanice analitycznej 
Recent publications
1.  Kovalchuk V., Mladenov I.M.^{♦}, Mechanics of infinitesimal gyroscopes on Mylar balloons and their actionangle analysis, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 01704214, DOI: 10.1002/mma.6099, pp.112, 2020 Abstract: Here, we apply the general scheme for description of mechanics of infinitesimal bodies in Riemannian spaces to the example of geodetic and nongeodetic (for two different model potentials) motions of infinitesimal rotators on the Mylar balloon. The structure of partial degeneracy is investigated with the help of the corresponding HamiltonJacobi equation and actionangle analysis. In all situations it was found that for any of the sixth disjoint regions in the phase space among the three action variables only two are essential for the description of our models at the level of the old quantum theory (according to the BohrSommerfeld postulates). Moreover, in both nongeodetic models, the action variables were intertwined with the quantum number N corresponding to the quantized value of the radius r of the inflated Mylar balloon. Keywords:actionangle analysis, affinelyrigid bodies, BohrSommerfeld quantum theory, HamiltonJacobi equation, infinitesimal gyroscopes, Mylar balloons, residue analysis. Affiliations:
 
2.  Kovalchuk V., Gołubowska B., Mladenov I.M.^{♦}, Mechanics of infinitesimal test bodies on Delaunay surfaces: spheres and cylinders as limits of unduloids and their actionangle analysis, Journal of Geometry and Symmetry in Physics, ISSN: 13125192, DOI: 10.7546/jgsp5320195584, Vol.53, pp.5584, 2019 Abstract: This paper discusses the motion of infinitesimal gyroscopes along the twodimensional surfaces of constant mean curvature embedded into the threedimensional Euclidean space. We have considered the cases of unduloids, spheres, and cylinders for which the corresponding HamiltonJacobi equations are written and analyzed with the help of the actionangle variables. Spheres and cylinders are considered as limiting cases of unduloids and the residue analysis was performed which provides the connection between all three action variables and the energy. This has been traced also for the geodetic situations and for two additional classical model potentials. Keywords:actionangle variables, affinelyrigid bodies, d'Alembert mechanics, Delaunay surfaces, geometry of curves and surfaces, HamiltonJacobi equation, harmonic/anharmonic oscillator potential models, infinitesimal test bodies, residue analysis, Riemannian spaces. Affiliations:
 
3.  Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Space‐time as a structured relativistic continuum, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 01704214, DOI: 10.1002/mma.5087, Vol.41, pp.54045422, 2018 Abstract: There are various models of gravitation: the metrical Hilbert‐Einstein theory, a wide class of intrinsically Lorentz‐invariant tetrad theories (generally covariant in the space‐time sense), and many gauge models based on various internal symmetry groups (Lorentz, Poincare, GL(n,R), SU(2,2), GL(4,C), etc). The gauge models are usually preferred but nevertheless it is an interesting idea to develop the class of GL(4,R)‐invariant (or rather GL(n,R)‐invariant) tetrad (n‐leg) generally covariant models. This is done below and motivated by our idea of bringing back to life the Thales of Miletus concept of affine symmetry. Formally, the obtained scheme is a generally covariant tetrad (n‐leg) model, but it turns out that generally covariant and intrinsically affinely invariant models must have a kind of nonaccidental Born‐Infeld‐like structure. Let us also mention that they, being based on tetrads (n‐legs), have many features common with continuous defect theories. It is interesting that they possess some group‐theoretical solutions and more general spherically symmetric solutions, discussion of which is the main new result presented in this paper, including the applications of the 't Hooft‐Polyakov monopoles in the generally covariant theories, which enables us to find some rigorous solutions of our strongly nonlinear equations. It is also interesting that within such a framework, the normal‐hyperbolic signature of the space‐time metric is not introduced by hand but appears as a kind of solution, rather integration constants, of differential equations. Let us mention that our Born‐Infeld scheme is more general than alternative tetrad models. It may be also used within more general schemes, including also the gauge ones. Keywords:micromorphic medium, modified gravity, relativistic continuum, spherically symmetric solutions, theory of fundamental interactions. Affiliations:
 
4.  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: 0022247X, DOI: 10.1016/j.jmaa.2016.08.042, Vol.446, pp.493520, 2017 Abstract: 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. Keywords:Homogeneous deformation, structured media, affinelyinvariant dynamics, elastic vibrations encoded in kinetic energy, CalogeroMoser and Sutherland integrable lattices Affiliations:
 
5.  Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Quantized mechanics of affinelyrigid bodies, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 01704214, DOI: 10.1002/mma.4501, Vol.40, No.18, pp.69006918, 2017 Abstract: 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 semidirect 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 2polar decomposition of the matrix of the internal configuration and on the PeterWeyl 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. Keywords:Homogeneously deformable body, PeterWeyl analysis, Schrödinger quantization. Affiliations:
 
6.  Gołubowska B., Kovalchuk V., Sławianowski J.J., Constraints and symmetry in mechanics of affine motion, JOURNAL OF GEOMETRY AND PHYSICS, ISSN: 03930440, DOI: 10.1016/j.geomphys.2014.01.012, Vol.78, pp.5979, 2014 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 twopolar decomposition of affine motion. Later on some additional constraints imposed on the affine motion are reviewed, both holonomic and nonholonomic. In particular, we concentrate on certain natural nonholonomic models of the rotationless 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 rotationless motion as a timedependent family of symmetric matrices of placements. The latter model seems to be nongeometric and nonphysical. Nevertheless, there are certain relationships between our nonholonomic models and the polar decomposition. Keywords:Affine motion, polar and twopolar decompositions, nonholonomic constraints, d’Alembert and Lusternik variational principles, vakonomic constraints Affiliations:
 
7.  Popov A.^{♦}, Kovalchuk V., Parametric representation of wave propagation in nonuniform media (both in transmission and stop bands), MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 01704214, DOI: 10.1002/mma.2687, Vol.36, No.11, pp.13501362, 2013 Abstract: An analytical approach based on the parametric representation of the wave propagation in nonuniform 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 quasiperiodic Floquet solutions was shown. Keywords:Floquet theorem, parametric resonance/antiresonance, wave propagation, nonuniform media, periodic structures, stop and transmission bands Affiliations:
 
8.  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 SYSTEMSSERIES B, ISSN: 15313492, DOI: 10.3934/dcdsb.2012.17.699, Vol.17, No.2, pp.699733, 2012 Abstract: 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 nonperturbative 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 BornInfeld nonlinearity. Some interesting, a bit surprising problems concerning Noether theorem are discussed, in particular in the context of large symmetry groups. Keywords:Essential nonlinearity, dynamical symmetries, affine bodies, general relativity, tetrad models and micromorphic continua Affiliations:
 
9.  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: 01704214, DOI: 10.1002/mma.1531, Vol.35, pp.1742, 2012 Abstract: 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’. Keywords:Weyl–Wigner–Moyal–Ville formalism, topological groups, classical momentum mapping, unitary projective representations, Pontryagin duality, ‘Umklapp–Prozessen’ Affiliations:
 
10.  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: 13125192, DOI: 10.7546/jgsp2620123359, Vol.26, pp.3359, 2012 Abstract: Discussed are certain strange properties of the Galilei group, connected first of all with the property of mechanical energymomentum 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 WeylWignerMoyalVille approach to quantum mechanics. Galilei group, affine transformation, particle mass, projective unitary representation, WeylWignerMoyalVille formalism Affiliations:
 
11.  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: 01704214, DOI: 10.1002/mma.1462, Vol.34, pp.15121540, 2011 Abstract: 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 affinemultibody theory is expected to provide a new class of micromorphic media. Keywords:Systems of affine bodies, mutual deformation tensors, affine and Euclidean invariance, structured media, elastic vibrations, geometric structures Affiliations:
 
12.  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 QuantumMechanical Models with Symmetries, Journal of Geometry and Symmetry, ISSN: 13125192, DOI: 10.7546/jgsp2120116194, Vol.21, pp.6194, 2011 Abstract: 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 groupalgebraic 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). Keywords:Systems of angular momenta, models with symmetries, quantum dynamics, spin systems, Schroedinger quantum mechanics, quasiclassical limit Affiliations:
 
13.  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, ISSN: 13125192, DOI: 10.7546/jgsp2220116794, Vol.22, pp.6794, 2011 Abstract: In Part I of this series we have presented the general ideas of applying groupalgebraic 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. Systems of angular momenta, models with symmetries, quantum dynamics, Lie group structures, quasiclassical representation, classical analogy Affiliations:
 
14.  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, ISSN: 13125192, DOI: 10.7546/jgsp2320115995, Vol.23, pp.5995, 2011 Abstract: 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. Keywords:Systems of angular momenta, models with symmetries, quantum dynamics, quasiclassical isospin problems Affiliations:
 
15.  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: 05874254, DOI: 10.5506/APhysPolB.41.165, Vol.41, No.1, pp.165218, 2010 Abstract: 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. Keywords:Internal degrees of freedom, isometric and affine invariance, classical and quantum excitations, multiparticle objects, Raman scattering, nuclear radiation Affiliations:
 
16.  Sławianowski J.J., Kovalchuk V., Schrödinger and related equations as Hamiltonian systems, manifolds of secondorder tensors and new ideas of nonlinearity in quantum mechanics, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 00344877, DOI: 10.1016/S00344877(10)00008X, Vol.65, No.1, pp.2976, 2010 Abstract: Considered is the Schrödinger equation in a finitedimensional space as an equation of mathematical physics derivable from the variational principle and treatable in terms of the LagrangeHamilton 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 secondorder time derivatives and introduce a kind of nondirect, nonperturbative, geometricallymotivated 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”. Keywords:Hamiltonian systems on manifolds of scalar products, finitelevel quantum systems, finitedimensional 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 Affiliations:
 
17.  Kovalchuk V., On classical dynamics of affinelyrigid bodies subject to the KirchhoffLove constraints, Symmetry, Integrability and Geometry: Methods and Applications SIGMA, ISSN: 18150659, DOI: 10.3842/SIGMA.2010.031, Vol.6, No.031, pp.112, 2010 Abstract: In this article we consider the affinelyrigid body moving in the threedimensional physical space and subject to the Kirchhoff–Love constraints, i.e., while it deforms homogeneously in the twodimensional central plane of the body it simultaneously performs onedimensional 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 twopolar (singular value) decomposition. Keywords:Affinelyrigid bodies with degenerate dimension, Kirchhoff–Love constraints, polar decomposition, Green deformation tensor, deformation invariants, stationary ellipsoids as special solutions Affiliations:
 
18.  Kovalchuk V., Rożko E.E., Classical models of affinelyrigid bodies with "thickness" in degenerate dimension, Journal of Geometry and Symmetry in Physics, ISSN: 13125192, DOI: 10.7546/jgsp1420095165, Vol.14, pp.5165, 2009 Abstract: 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 mdimensional affinelyrigid body moving in the ndimensional physical space, m < n. We mainly concentrate on the physical situation m = 2, n = 3 when “thickness” of flat bodies performs onedimensional oscillations orthogonal to the twodimensional 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. Keywords:Affinelyrigid bodies with degenerate dimension, nonzeroth thickness, twopolar decomposition, stationary ellipsoidal figures of equilibrium Affiliations:
 
19.  Kovalchuk V., Sławianowski J.J., Hamiltonian systems inspired by the Schrödinger equation, Symmetry, Integrability and Geometry: Methods and Applications SIGMA, ISSN: 18150659, DOI: 10.3842/SIGMA.2008.046, Vol.4, pp.4654, 2008 Abstract: Described is nlevel quantum system realized in the ndimensional “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 secondorder modified Schroedinger equations are obtained. Keywords:Schroedinger equation, Hamiltonian systems on manifolds of scalar products, nlevel quantum systems, scalar product as a dynamical variable, essential nonperturbative nonlinearity, conservation laws, GL_n(C)invariance Affiliations:
 
20.  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: 00344877, DOI: 10.1016/S00344877(05)800023, Vol.55, No.1, pp.146, 2005 Abstract: 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 manybody problems are suggested. Discussed is also the possibility of halfinteger angular momentum in composed systems of spinless particles. Keywords:Collective modes, affine invariance, Schroedinger quantization, quantum manybody problem Affiliations:
 
21.  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: 00344877, DOI: 10.1016/S00344877(04)800260, Vol.54, No.3, pp.373427, 2004 Abstract: 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 affinelyinvariant. The relationship with the dynamics of integrable onedimensional lattices is discussed. It is shown that affinelyinvariant geodetic models may encode the dynamics of something like elastic vibrations. Keywords:Collective modes, affine invariance, integrable lattices, nonlinear elasticity Affiliations:
 
22.  Sławianowski J.J., Kovalchuk V., Classical and quantized affine physics: a step towards it, Journal of Nonlinear Mathematical Physics, ISSN: 14029251, DOI: 10.2991/jnmp.2004.11.s1.21, Vol.11, No.Supplement, pp.157166, 2004 Abstract: 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 (affinelyrigid bodies). The special attention is paid to left and rightinvariant 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. Keywords:Lie groups, homogeneous spaces, defomable bodies, left and right affine invariance, geodetic models, classical and quantum mechanics Affiliations:
 
23.  Kovalchuk V., Green function for KleinGordonDirac equation, Journal of Nonlinear Mathematical Physics, ISSN: 14029251, DOI: 10.2991/jnmp.2004.11.s1.9, Vol.11, No.Supplement, pp.7277, 2004 Abstract: The Green function for KleinGordonDirac equation is obtained. The case with the dominating KleinGordon term is considered. There seems to be a formal analogy between our problem and a certain problem for a 4dimensional 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 Texponent formalism. Keywords:Green functions, KleinGordonDirac equation, Texponent formalism, Dirac plane harmonic waves Affiliations:
 
24.  Sławianowski J.J., Kovalchuk V., Invariant geodetic problems on the affine group and related Hamiltonian systems, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 00344877, DOI: 10.1016/S00344877(03)800290, Vol.51, No.2/3, pp.371379, 2003 Abstract: 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 nongeodetic problems in corresponding Riemannian spaces are discussed. Keywords:Affinelyrigid body, Riemannian metrics, geodetic problems, twopolar decomposition, Hamiltonian systems Affiliations:
 
25.  Sławianowski J.J., Kovalchuk V., KleinGordonDirac equation: physical justification and quantization attempts, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 00344877, DOI: 10.1016/S00344877(02)800234, Vol.49, No.2/3, pp.249257, 2002 Abstract: Discussed is the KleinGordonDirac 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 energymomentum tensor and 4current. It appears, in particular, that the scalar product is a superposition of KleinGordon 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. Keywords:KleinGordonDirac equation, plane harmonic waves with different masses, Dirac bispinors, Lagrange formalism, canonical formalism Affiliations:

List of recent monographs
1. 67  Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Dynamical systems with internal degrees of freedom in nonEuclidean spaces, IPPT Reports on Fundamental Technological Research, 8, pp.1129, 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.1164, 2004 
List of chapters in recent monographs
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.335, 2015  
2. 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.267279, 2015  
3. 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.81103, 2013  
4. 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.150175, 2012  
5. 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.70155, 2011  
6. 147  Kovalchuk V., Rożko E.E., Geometry, integrability and quantization, Proceedings of the 10th International Conference, rozdział: Classical models of affinelyrigid bodies with thickness in degenerate dimension, Bulgarian Academy of Sciences, Avangard Prima, Mladenov I.M., Vilasi G., Yoshioka A. (Eds.), pp.197210, 2009  
7. 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.119150, 2009  
8. 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.63132, 2008 
Editor of monographs
1. 385  Kovalchuk V., Geometry, Integrability, Mechanics and Quantization, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.1461, 2015 
Conference papers
1.  Sławianowski J.J., Kovalchuk V., Quantized version of the theory of affine body, Geometry, Integrability and Quantization, ISSN: 13143247, DOI: 10.7546/giq1620157393, Vol.XVI, pp.7393, 2015 Abstract: In the previous lecture we have introduce and discussed the concept of affinelyrigid, 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 highlysymmetric, in particular affinelyinvariant, models and the PeterWeyl analysis of wave functions. Keywords:Homogeneously deformable body, Schroedinger quantization, affine invariance, highly symmetric models, PeterWeyl analysis Affiliations:
 
2.  Kovalchuk V., On new ideas of nonlinearity in quantum mechanics, Geometry, Integrability and Quantization, ISSN: 13143247, DOI: 10.7546/giq162015195206, Vol.XVI, pp.195206, 2015 Abstract: Our main idea is to suggest a new model of nonperturbative and geometrically motivated nonlinearity in quantum mechanics. The Schroedinger equation and corresponding relativistic linear wave equations derivable from variational principles are analyzed as usual selfadjoint equations of mathematical physics. It turns out that introducing the secondorder 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 Hilbertspace 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 highsymmetry problems. Keywords:Nonperturbative nonlinearity, selfadjoint dynamical equations, Schroedinger equation, highly symmetric problems, Hilbertspace scalar product Affiliations:
 
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: 13143247, DOI: 10.7546/gip142013103115, Vol.XIV, pp.103115, 2013 Abstract: 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. Keywords:Affine motion, nonholonomic constraints, d’Alembert and Lusternik variational principles, vakonomic constraints Affiliations:
 
4.  Sławianowski J.J., Kovalchuk V., Invariant geodetic problems on the projective group Pr(n,R), 5th International Conference “Symmetry in NonlinearMathematical Physics”, 20030623/0629, Kiev (UA), Vol.50, No.2, pp.955960, 2004 Abstract: The concept of ndimensional projectivelyrigid body is introduced and its connection to the concept of (n+ 1)dimensional incompressible affinelyrigid body is analysed. The equations of geodetic motion for such a projectivelyrigid body are obtained. As an instructive example, the special case of n = 1 is investigated. Keywords:Projectivelyrigid bodies, incompressible affinelyrigid bodies, left and right invariant problems, geodetic motion Affiliations:
