Ewa Rożko, Ph.D. 

Doctoral thesis
20070220  Układy dynamiczne na przestrzeniach jednorodnych i ich zastosowanie w mechanice kontinuum
 599 
Recent publications
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: 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:
 
2.  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, pp.119, 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:
 
3.  Rożko E.E., Gobcewicz E., Quantization of systems with internal degrees of freedom in twodimensional manifolds, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 00344877, DOI: 10.1016/S00344877(14)600483, Vol.73, No.3, pp.325343, 2014 Abstract: Presented is a primary step towards quantization of infinitesimal rigid body moving in a twodimensional manifold. The special stress is laid on spaces of constant curvature like the twodimensional sphere and pseudosphere (Lobachevsky space). Also twodimensional torus is briefly discussed as an interesting algebraic manifold. Keywords:quantization, infinitesimal rigid body, Lobachevsky space, the Schrödinger procedure Affiliations:
 
4.  Rożko E.E., Sławianowski J.J., Essential Nonlinearity in Field Theory and Continuum Mechanics. Second and FirstOrder GenerallyCovariant Models, Journal of Geometry and Symmetry in Physics, ISSN: 13125192, DOI: 10.7546/jgsp3420145176, Vol.34, pp.5176, 2014 Abstract: Discussed is the problem of the mutual relationship of differentially firstordr and secondorder field theories and quantummechanical concepts. We show that unlike the real history of physics, the theories with algebraically secondorder Lagrangians are primary, and in case more adequate. It is shown that in principle, the Schodinger idea about Lagrangians which are quadratic in derivatives, and leading to secondorder differential equations, is not only acceptable, but just it opens some new perspective in field theory. This has to do with using the Lorentzconformal or rather its universal covering SU(2,3) as a gauge group. This has also some influence on the theory of defect in continua. Keywords:field theory, gauge group, Lagrangians Affiliations:
 
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 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:
 
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: 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:
 
7.  Sławianowski J.J., Gołubowska B., Rożko E.E., SO(4,R), related groups and threedimensional twogyroscopic problems, ACTA PHYSICA POLONICA B, ISSN: 05874254, DOI: 10.5506/APhysPolB.43.19, Vol.43, No.1, pp.1949, 2012 Abstract: Discussed are some problems of two (or more) mutually coupled systems with gyroscopic degrees of freedom. First of all, we mean the motion of a small gyroscope in the nonrelativistic Einstein Universe RXS3(0;R); the second factor denoting the Euclidean 3sphere of radius R in R4. But certain problems concerning twogyroscopic systems in Euclidean space R3 are also mentioned. The special stress is laid on the relationship between various models of the configuration space like, e.g., SU(2)xSU(2), SO(4;R), SO(3;R)xSO(3;R) etc. They are locally diffeomorphic, but globally different. We concentrate on classical problems, nevertheless, some quantum aspects are also mentioned. Keywords:Rigid body, gyroscopic degrees of freedom, Einstein Universe, Euclidean space, SU(2)xSU(2), SO(4;R), SO(3;R)xSO(3;R) Affiliations:
 
8.  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:
 
9.  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:
 
10.  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 in Physics, 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:
 
11.  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/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:
 
12.  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/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:
 
13.  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:
 
14.  Rożko E.E., Quantization of affinelyrigid bodies with degenerate dimension, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 00344877, DOI: 10.1016/S00344877(10)000078, Vol.65, No.1, pp.115, 2010 Abstract: 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 3dimensional situation and on the dynamical models isotropic both in space and material. Basing on the PeterWeyl 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. Keywords:affinelyrigid bodies, affine injections, homogeneous spaces, Schrodinger quantization, deformable rotors quantized Affiliations:
 
15.  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:
 
16.  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:
 
17.  Rożko E.E., Dynamics of affinelyrigid bodies with degenerate dimension, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 00344877, DOI: 10.1016/S00344877(05)800874, Vol.56, No.3, pp.311332, 2005 Abstract: 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 ℝm into ℝn. 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. Keywords:affinelyrigid bodies, linear groups, homogenous spaces, stationary solutions, Grassmann, Stiefel manifolds Affiliations:
 
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: 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:

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. 436  Sławianowski J.J., Rożko E.E., Continuous Media with Microstructure 2, rozdział: Affinely Rigid Body and Affine Invariance in Physics, Springer International Publishing, Switzerland, Bettina Albers and Mieczysław Kuczma (Eds.), pp.95118, 2016 
2. 390  Sławianowski J.J., Rożko E.E., Geometry, Integrability, Mechanics and Quantization, rozdział: Classical and quantization problems in degenerate affine motion, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.135159, 2015 
3. 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 
4. 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 
5. 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 
Conference papers
1.  Sławianowski J.J., Rożko E.E., Classical and quantization problems in degenerate affine motion, Geometry, Integrability and Quantization, ISSN: 13143247, DOI: 10.7546/giq162015139163, Vol.XVI, pp.139163, 2015 Abstract: Discussed are classical and quantized models of affinely rigid motion with degenerate dimension, i.e., such ones that the geometric dimensions of the material and physical spaces need not be equal to each other. More precisely, the material space may have dimension lower than the physical space. Physically interesting are special cases m = 2 or m = 1 and n = 3, first of all m = 2, n = 3, i.e., roughly speaking, the affinely deformable coin in three–dimensional Euclidean space. We introduce some special coordinate systems generalizing the polar and two–polar decompositions in the regular case. This enables us to reduce the dynamics to two degrees of freedom. In quantum case this is the reduction of the Schrödinger equation to multicom ponent wave functions of two deformation invariants. Keywords:affinely rigid motion, degenerate dimension, polar and twopolar decompositions, multicomponent wave function, Schrödinger equation, quantized models, deformation invariants Affiliations:
 
2.  Rożko E.E., Sławianowski J.J., Essential nonlinearity in field theory and continuum mechanics. Second and firstorder generallycovariant models, Geometry, Integrability and Quantization, ISSN: 13143247, DOI: 10.7546/giq152014218241, Vol.XV, pp.218241, 2014 Abstract: Discussed is the problem of the mutual relationship of differentially firstordr and secondorder field theories and quantummechanical concepts. We show that unlike the real history of physics, the theories with algebraically secondorder Lagrangians are primary, and in case more adequate. It is shown that in principle, the Schodinger idea about Lagrangians which are quadratic in derivatives, and leading to secondorder differential equations, is not only acceptable, but just it opens some new perspective in field theory. This has to do with using the Lorentzconformal or rather its universal covering SU(2,3) as a gauge group. This has also some influence on the theory of defect in continua. Keywords:field theory, gauge group, Lagrangians 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:
