Prof. Jan Sławianowski, Ph.D., Dr. Habil.

Department of Theory of Continuous Media and Nanostructures (ZTOCiN)
Analytical Mechanics and Field Theory (PMAiTP)
retiree
telephone: (+48) 22 826 12 81 ext.: 135
room: 114
e-mail: jslawian
personal site: http://bluebox.ippt.pan.pl/~jslawian

Habilitation thesis
1976Teoria deformacji wielomianowych806
 
Professor
1988Title of professor
Supervision of doctoral theses
1.2007-02-20Rożko Ewa  Układy dynamiczne na przestrzeniach jednorodnych i ich zastosowanie w mechanice kontinuum599
 
2.2006-06-29Martens Agnieszka Hamiltonowskie i kwantowe układy z symetriami i więzami. Modele nieliniowe i ich zastosowania fizyczne596
 
3.2006-06-29Gołubowska Barbara Ciało afinicznie sztywne w zakrzywionych przestrzeniach i rozmaitościach o stałej krzywiźnie597
 
4.2006-03-31Kovalchuk Vasyl Nonlinear models of collective and internal degrees of freedom in mechanics and field theory591
 
5.1996-04-27Godlewski Piotr  Uogólnione nieliniowości typu Borna-Infelda w mechanice i teorii pola 
6.1994-06-30Makaruk Hanna  Grupowe i pseudogrupowe modele czasoprzestrzeni 
7.1990Dorosiewicz Sławomir  O niektórych zagadnieniach teorii układów dynamicznych na rozmaitościach skończenie wymiarowych 
8.1981Rudnicki Marek  Dyskretny model ośrodka z mikrostrukturą 
9.1980Żuchowski Krzysztof  Fluktuacje elektrodynamiczne i reakcja ośrodka przewodzącego na zaburzenie zewnętrzne 
10.1979Seredyńska Małgorzata  Drgania nieliniowe ciał afinicznie sztywnych 

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: 0022-247X, DOI: 10.1016/j.jmaa.2016.08.042, Vol.446, pp.493-520, 2017
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

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

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
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

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 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.

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

3.Ali G., Beneduci R., Mascali G., Schroeck Jr. F.E., Sławianowski J.J., Some Mathematical Considerations on Solid State Physics in the Framework of the Phase Space Formulation of Quantum Mechanics, INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS, ISSN: 0020-7748, DOI: 10.1007/s10773-013-1912-9, Vol.53, pp.3546-3574, 2014
Ali G., Beneduci R., Mascali G., Schroeck Jr. F.E., Sławianowski J.J., Some Mathematical Considerations on Solid State Physics in the Framework of the Phase Space Formulation of Quantum Mechanics, INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS, ISSN: 0020-7748, DOI: 10.1007/s10773-013-1912-9, Vol.53, pp.3546-3574, 2014

Abstract:
We propose, for a finite crystal, that the entire system be expressed in terms of the phase space, including momentum, configuration position and spin. This is to be done both classically and quantum mechanically. We illustrate this with a silicon crystal. Then, quantum mechanically measuring with a wire containing electrons, we obtain a theoretically good approximation for an electron in one semiconductor. As well, we outline what has to be done for any crystal.

Keywords:
Quantum mechanics in Phase space, Symmetry groups for a crystal, Groups representation, Quantization

4.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
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

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 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.

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

5.Rożko E.E., Sławianowski J.J., Essential Nonlinearity in Field Theory and Continuum Mechanics. Second- and First-Order Generally-Covariant Models, Journal of Geometry and Symmetry in Physics, ISSN: 1312-5192, DOI: 10.7546/jgsp-34-2014-51-76, Vol.34, pp.51-76, 2014
Rożko E.E., Sławianowski J.J., Essential Nonlinearity in Field Theory and Continuum Mechanics. Second- and First-Order Generally-Covariant Models, Journal of Geometry and Symmetry in Physics, ISSN: 1312-5192, DOI: 10.7546/jgsp-34-2014-51-76, Vol.34, pp.51-76, 2014

Abstract:
Discussed is the problem of the mutual relationship of differentially first-ordr and second-order field theories and quantum-mechanical concepts. We show that unlike the real history of physics, the theories with algebraically second-order 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 second-order differential equations, is not only acceptable, but just it opens some new perspective in field theory. This has to do with using the Lorentz-conformal 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

6.Blackmore D., Prykarpatski A.K., Bogolubov Jr. N.N., Sławianowski J.J., Mathematical foundations of the classical Maxwell-Lorentz electrodynamic models in the canonical Lagrangian and Hamiltonian formalisms, Universal Journal of Physics and Application, ISSN: 2331-6535, DOI: 10.13189/ujpa.2013.010216, Vol.1, No.2, pp.160-178, 2013
Blackmore D., Prykarpatski A.K., Bogolubov Jr. N.N., Sławianowski J.J., Mathematical foundations of the classical Maxwell-Lorentz electrodynamic models in the canonical Lagrangian and Hamiltonian formalisms, Universal Journal of Physics and Application, ISSN: 2331-6535, DOI: 10.13189/ujpa.2013.010216, Vol.1, No.2, pp.160-178, 2013

Abstract:
We present new mathematical foundations of classical Maxwell–Lorentz electrodynamic models and related charged particles interaction-radiation problems, and analyze the fundamental least action principles via canonical Lagrangian and Hamiltonian formalisms. The corresponding electrodynamic vacuum field theory aspects of the classical Maxwell–Lorentz theory are analyzed in detail. Electrodynamic models of charged point particle dynamics based on a Maxwell type vacuum field medium description are described, and new field theory concepts related to the mass particle paradigms are discussed. We also revisit and reanalyze the mathematical structure of the classical Lorentz force expression with respect to arbitrary inertial reference frames and present new interpretations of some classical special relativity theory relationships.

Keywords:
Lagrangian and Hamiltonian Formalism, Least Action Principle, Vacuum Field Theory, Lorentz Force Problem, Feynman Approach, Maxwell Equations, Lorentz Constraint, Electron Radiation Theory

7.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
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

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 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.

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

8.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
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

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’

9.Sławianowski J.J., Gołubowska B., Rożko E.E., SO(4,R), related groups and three-dimensional two-gyroscopic problems, ACTA PHYSICA POLONICA B, ISSN: 0587-4254, DOI: 10.5506/APhysPolB.43.19, Vol.43, No.1, pp.19-49, 2012
Sławianowski J.J., Gołubowska B., Rożko E.E., SO(4,R), related groups and three-dimensional two-gyroscopic problems, ACTA PHYSICA POLONICA B, ISSN: 0587-4254, DOI: 10.5506/APhysPolB.43.19, Vol.43, No.1, pp.19-49, 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 non-relativistic Einstein Universe RXS3(0;R); the second factor denoting the Euclidean 3-sphere of radius R in R4. But certain problems concerning two-gyroscopic 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)

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: 1312-5192, DOI: 10.7546/jgsp-26-2012-33-59, Vol.26, pp.33-59, 2012
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

Abstract:
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.

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

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: 0170-4214, DOI: 10.1002/mma.1462, Vol.34, pp.1512-1540, 2011
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

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 affine-multibody 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

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 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
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

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 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).

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

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 in Physics, DOI: 10.7546/jgsp-22-2011-67-94, Vol.22, pp.67-94, 2011
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

Abstract:
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.

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

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 in Physics, DOI: 10.7546/jgsp-23-2011-59-95, Vol.23, pp.59-95, 2011
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

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

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: 0587-4254, DOI: 10.5506/APhysPolB.41.165, Vol.41, No.1, pp.165-218, 2010
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

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

16.Martens A., Sławianowski J.J., Affinely-rigid body and oscillatory dynamical models on GL(2,R), ACTA PHYSICA POLONICA B, ISSN: 0587-4254, Vol.41, No.8, pp.1847-1880, 2010
Martens A., Sławianowski J.J., Affinely-rigid body and oscillatory dynamical models on GL(2,R), ACTA PHYSICA POLONICA B, ISSN: 0587-4254, Vol.41, No.8, pp.1847-1880, 2010

Abstract:
Discussed is a model of the two-dimensional affinely-rigid body with the double dynamical isotropy. We investigate the systems with potential energies for which the variables can be separated. The special stress is laid on the model of the harmonic oscillator potential and certain anharmonic alternatives. Some explicit solutions are found on the classical, quasiclassical (Bohr–Sommerfeld) and quantum levels.

Keywords:
affinely-rigid body, harmonic oscillator potential, Sommerfeld polynomial method

17.Sławianowski J.J., Geometric nonlinearities in field theory, condensed matter and analytical mechanics, CONDENSED MATTER PHYSICS, ISSN: 1607-324X, Vol.13, No.4, pp.43103:1-19, 2010
18.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
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

Abstract:
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”.

Keywords:
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

19.Sławianowski J.J., Gołubowska B., Motion of test bodies with internal degrees of freedom in non-euclidean spaces, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, Vol.65, No.3, pp.379-422, 2010
Sławianowski J.J., Gołubowska B., Motion of test bodies with internal degrees of freedom in non-euclidean spaces, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, Vol.65, No.3, pp.379-422, 2010

Abstract:
Discussed is mechanics of objects with internal degrees of freedom in generally non- Euclidean spaces. Geometric peculiarities of the model are investigated in detail. Discussed are also possible mechanical applications, e.g. in dynamics of structured continua, defect theory and in other fields of mechanics of deformable bodies. Elaborated is a new method of analysis based on nonholonomic frames. We compare our results and methods with those of other authors working in nonlinear dynamics. Simple examples are presented.

Keywords:
affine invariance, affinely-rigid bodies, collective modes, internal degrees of freedom, nonlinear elasticity, Riemannian manifolds

20.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
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

Abstract:
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.

Keywords:
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

21.Burov A.A., Motte I., Sławianowski J.J., Stepanov S.Y., Zadachi issledovanija ustojchivosti i stabilizatsii dvizhenija, Vychislitel’nyj Tsentr Rossijskoj Akademii Nauk, pp.93-106, 2006
22.Sławianowski J.J., Quantum systems on linear groups, INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS, ISSN: 0020-7748, Vol.44, No.11, pp.2027-2037, 2005
23.Sławianowski J.J., Classical and quantized affine models of structured media, MECCANICA, ISSN: 0025-6455, Vol.40, pp.365-387, 2005
24.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
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

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 many-body problems are suggested. Discussed is also the possibility of half-integer angular momentum in composed systems of spinless particles.

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

25.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
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

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 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.

Keywords:
Collective modes, affine invariance, integrable lattices, nonlinear elasticity

26.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
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

Abstract:
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.

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

27.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
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

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 (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.

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

28.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
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

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 non-geodetic problems in corresponding Riemannian spaces are discussed.

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

29.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
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

Abstract:
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.

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

30.Sławianowski J.J., Generally-covariant Field Theories and Space-time as a Micromorphic Continuum, Prace IPPT - IFTR Reports, ISSN: 2299-3657, No.51, pp.1-92, 1988
31.Sławianowski J.J., Teoria deformacji wielomianowych.(Praca habilitacyjna), Prace IPPT - IFTR Reports, ISSN: 2299-3657, No.45, pp.1-104, 1975
32.Sławianowski J.J., Mechanika analityczna deformacji jednorodnych, Prace IPPT - IFTR Reports, ISSN: 2299-3657, No.8, pp.1-38, 1973
33.Sławianowski J.J., Informacja a symetria rozkładów w mechanice statystycznej. - Klasyczne rozkłady czyste, Prace IPPT - IFTR Reports, ISSN: 2299-3657, No.30, pp.1-19, 1972
34.Gołębiowska A.A., Sławianowski J.J., Geometria przestrzeni fazowej, Prace IPPT - IFTR Reports, ISSN: 2299-3657, No.28, pp.1-40, 1971

List of recent monographs
1.
473
Sławianowski J.J., Schroeck Jr. F.E., Martens A., Why must we work in the phase space?, IPPT Reports on Fundamental Technological Research, 1, pp.1-162, 2016
2.
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
3.
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
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.95-118, 2016
2.
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
3.
386
Sławianowski J.J., Geometry, Integrability, Mechanics and Quantization, rozdział: The two apparently different but hiddenly related Euler achievements: rigid body and ideal fluid. Our unifying going between: affinely-rigid body and affine invariance in physics, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.31-67, 2015
4.
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
5.
388
Sławianowski J.J., Martens A., Geometry, Integrability, Mechanics and Quantization, rozdział: Affinely-rigid body and oscillatory two-dimensional models, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.89-105, 2015
6.
389
Sławianowski J.J., Gołubowska B., Geometry, Integrability, Mechanics and Quantization, rozdział: Bertrand systems on spaces of constant sectional curvature. The action-angle analysis. Classical, quasi-classical and quantum problems, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.106-134, 2015
7.
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.135-159, 2015
8.
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
9.
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
10.
303
Vassilev V.M., Djondjorov P.A., Hadzhilazova M.T., Mladenov I.M., Sławianowski J.J., Mechanics and Nanomaterials and Nanotechnology, Series in Applied Mathematics, rozdział: Equilibrium shapes of fluid membranes and carbon nanostructures, Institute of Mechanics Bulgarian Academy of Sciences, 3, pp.153-184, 2012
11.
292
Sławianowski J.J., Classical and Celestial Mechanics, Selected Papers, rozdział: Systems of Hamilton-Jacobi Equations in Terms of Symplectic and Contact Geometry, Wydawnictwo Collegium Mazovia, Russian Academy of Science, Lomonosov Moscow State University, Moscow State Aviation Institute, Dorodnitsyn Computing Centre of RAS, Collegium Mazovia, Siedlce, L. Gadomski, P. Krasilnikov, and A. Prokopenya (Eds.), pp.170-193, 2012
12.
293
Sławianowski J.J., Gołubowska B., Classical and Celestial Mechanics, Selected Papers, rozdział: Hamiltonian Systems on Matrix Manifolds and Their Applications, Wydawnictwo Collegium Mazovia, Russian Academy of Science, Lomonosov Moscow State University, Moscow State Aviation Institute, Dorodnitsyn Computing Centre of RAS, Collegium Mazovia, Siedlce, L. Gadomski, P. Krasilnikov, and A. Prokopenya (Eds.), pp.158-169, 2012
13.
304
Sławianowski J.J., Mesurements in Quantum Mechanics, rozdział: Order of time derivatives in quantum-mechanical equations, InTech Europe, pp.57-74, 2012
14.
60
Sławianowski J.J., Mathematical Methods in Continuum Mechanics, A Series of Monographs Technical University of Łódź, rozdział: Symmetries and Constraints in Mechanics of Continua, Technical University of Łódź Press, K. Wilmański, B. Michalak, J. Jędrysiak (Eds.), pp.195-211, 2011
15.
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
16.
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
17.
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
18.
148
Sławianowski J.J., Geometry, integrability and quantization, Proceedings of the 8th International Conference, rozdział: Geometrically implied nonlinearities in mechanics and field theory, Bulgarian Academy of Sciences, Mladenov M.I., De Leon M. (Eds.), pp.48-118, 2007
19.
168
Sławianowski J.J., Material substructures in complex bodies, from atomic level to continuum, rozdział: Quantization of affine bodies: Theory and applications in mechanics of structured media, Elsevier, Capriz G., Mariano P.P. (Eds.), pp.80-162, 2007
20.
210
Sławianowski J.J., Topics in mathematical physics, General relativity and cosmology in honor of Jerzy Plebanski, rozdział: Teleparallelism, Modified Born-Infeld nonlinearity and space-time as a micromorphic Ether, World Scientific, García-Compeán H., Mielnik B., Montesinos M., Przanowski M. (Eds.), pp.441-452, 2006

Conference papers
1.Sławianowski J.J., The two apparently different but hiddenly related Euler achievements: rigid body and ideal fluid. Our unifying going between: affinely-rigid body and affine invariance in physics, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-36-72, Vol.XVI, pp.36-72, 2015
Sławianowski J.J., The two apparently different but hiddenly related Euler achievements: rigid body and ideal fluid. Our unifying going between: affinely-rigid body and affine invariance in physics, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-36-72, Vol.XVI, pp.36-72, 2015

Abstract:
Reviewed are ideas underlying our concept of affinely-rigid body. We do this from the perspective of the Leonhard Euler two main achieve- ments, known under his name: the mechanics of rigid body and the dynamics of incompressible ideal fluid. But we formulate the theory which is some-how placed between those two models. Our scheme is a finite-dimensional dynamical system, but admitting deformative degrees of freedom. We also stress the connection with the general idea of affine invariance in physics.

Keywords:
igid body, incompressible ideal fluid, affine invariance, finite-dimensional dynamical systems, deformative degrees of freedom

2.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
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

Abstract:
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.

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

3.Sławianowski J.J., Martens A., Affinely-rigid body and oscillatory two-dimensional models, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-94-109, Vol.XVI, pp.94-109, 2015
Sławianowski J.J., Martens A., Affinely-rigid body and oscillatory two-dimensional models, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-94-109, Vol.XVI, pp.94-109, 2015

Abstract:
Discussed are some classical and quantization problems of the affinely-rigid body in two dimensions. Strictly speaking, we consider the model of the harmonic oscillator potential and then discuss some natural anharmonic modifications. It is interesting that the considered doubly-isotropic models admit coordinate systems in which the classical and Schrödinger equations are separable and in principle solvable in terms of special functions on groups.

Keywords:
affinely-rigid body, quantization problems, two-dimensional models, harmonic oscillator potential, anharmonic modifications, doubly-isotropic models, Schrödinger equation, special functions, separability problem

4.Sławianowski J.J., Gołubowska B., Bertrand systems on spaces of constant sectional curvature. The action-angle analysis. Classical, quasi-classical and quantum problems, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-110-138, Vol.XVI, pp.110-138, 2015
Sławianowski J.J., Gołubowska B., Bertrand systems on spaces of constant sectional curvature. The action-angle analysis. Classical, quasi-classical and quantum problems, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-110-138, Vol.XVI, pp.110-138, 2015

Abstract:
Studied is the problem of degeneracy of mechanical systems the configuration space of which is the three-dimensional sphere, the elliptic space, i.e., the quotient of that sphere modulo the antipodal identification, and finally, the three-dimensional pseudo-sphere, namely, the Lobatchevski space. In other words, discussed are systems on groups SU(2), SO(3,R), and SL(2,R) or its quotient SO(1,2) . The main subject are completely degenerate Bertrand-like systems. We present the action-angle classical description, the corresponding quasi-classical analysis and the rigorous quantum formulas. It is interesting that both the classical action-angle formulas and the rigorous quantum mechanical energy levels are superpositions of the flat-space expression, with those describing free geodetic motion on groups.

Keywords:
action-angle description, Bertrand systems, completely degenerate problems, elliptic space, Lobatchevski space, quasi-classical analysis, sphere

5.Sławianowski J.J., Rożko E.E., Classical and quantization problems in degenerate affine motion, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-139-163, Vol.XVI, pp.139-163, 2015
Sławianowski J.J., Rożko E.E., Classical and quantization problems in degenerate affine motion, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-139-163, Vol.XVI, pp.139-163, 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 two-polar decompositions, multicomponent wave function, Schrödinger equation, quantized models, deformation invariants

6.Rożko E.E., Sławianowski J.J., Essential nonlinearity in field theory and continuum mechanics. Second- and first-order generally-covariant models, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-15-2014-218-241, Vol.XV, pp.218-241, 2014
Rożko E.E., Sławianowski J.J., Essential nonlinearity in field theory and continuum mechanics. Second- and first-order generally-covariant models, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-15-2014-218-241, Vol.XV, pp.218-241, 2014

Abstract:
Discussed is the problem of the mutual relationship of differentially first-ordr and second-order field theories and quantum-mechanical concepts. We show that unlike the real history of physics, the theories with algebraically second-order 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 second-order differential equations, is not only acceptable, but just it opens some new perspective in field theory. This has to do with using the Lorentz-conformal 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

7.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
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

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

8.Sławianowski J.J., Martens A., The dynamics of the field of linear frames and gauge gravitation, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/gip-14-2013-201-214, Vol.XIV, pp.201-214, 2013
Sławianowski J.J., Martens A., The dynamics of the field of linear frames and gauge gravitation, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/gip-14-2013-201-214, Vol.XIV, pp.201-214, 2013

Abstract:
The paper is motivated by gauge theories of gravitation and condensed matter, tetrad models of gravitation and generalized Born-Infeld type nonlinearity. The main idea is that any generally-covariant and GL(n,R)-invariant theory of the n-leg field (tetrad field when n=4) must have the Born-Infeld structure. This means that Lagrangian is given by the square root of the determinant of some second-order twice covariant tensor built in a quadratic way of the field derivatives. It is shown taht there exist interesting solutions of the group-theoretical structure. Some models of the interaction between gravitation and matter are suggested. It turns out that in a sense the space-time dimension n=4, the normal-hyperbolic signature and velocity of light are integration constants of our differential equations.

Keywords:
gauge gravitation, linear frames, condensed matter

9.Sławianowski J.J., Systems of Hamiltonian-Jacobi equations in terms of symplectic and contact geometry, 7th International Symposium on Classical and Celestial Mechanics, 2012-10-23/10-28, Siedlce (PL), pp.170-193, 2012
10.Sławianowski J.J., Gołubowska B., Hamiltonian systems on matrix manifolds and their physical applications, 7th International Symposium on Classical and Celestial Mechanics, 2012-10-23/10-28, Siedlce (PL), pp.158-169, 2012
Sławianowski J.J., Gołubowska B., Hamiltonian systems on matrix manifolds and their physical applications, 7th International Symposium on Classical and Celestial Mechanics, 2012-10-23/10-28, Siedlce (PL), pp.158-169, 2012

Abstract:
Schrödinger equation as a self-adjoint differential equation of mathematical physics is discussed. For simplicity, a finite-level system is considered. A modified Schrödinger equation with the second time derivatives is described and some direct nonlinearity is admitted. The key of our idea is the assumption that the scalar product is not fixed once for all, but is a dynamical quantity mutually interacting with the state vector. We assume that the Lagrangian term describing its dynamics has the large symmetry group, the total complex linear group. This implies the strong essential nonlinearity. There is a hope that this geometrically implied nonlinearity may explain the decoherence and measurement paradoxes in quantum mechanics.

Keywords:
Schrödinger equation, Krawietz-type matrices, affine rigid body