Barbara Gołubowska, Ph.D.

Department of Theory of Continuous Media and Nanostructures (ZTOCiN)
Analytical Mechanics and Field Theory (PMAiTP)
position: senior specialist
telephone: (+48) 22 826 12 81 ext.: 430
room: 117
e-mail: bgolub

Doctoral thesis
2006-06-29Ciało afinicznie sztywne w zakrzywionych przestrzeniach i rozmaitościach o stałej krzywiźnie 
supervisor -- Prof. Jan Sławianowski, Ph.D., Dr. Habil., IPPT PAN
597
 
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
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

Affiliations:
Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
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
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.

Affiliations:
Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
3.Gołubowska B., Some aspects of affine motion and nonholonomic constraints. Two ways to describe homogeneously deformable bodies, ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK, ISSN: 0044-2267, DOI: 10.1002/zamm.201400192, Vol.96, No.8, pp.968-985, 2016
Abstract:

This paper has been inspired by ideas presented by V. V. Kozlov in his works [23, 24]. In the present work the main goal is to carry out a thorough analysis of some geometric problems of the dynamics of affinely-rigid bodies. We present two ways to describe this case: the classical dynamical d'Alembert and variational, i.e., vakonomic ones. So far, we can see that they give quite different results. The vakonomic model from the mathematical point of view seems to be more elegant. The similar problems were examined by M. Jóźwikowski and W. Respondek in their paper [20].

Keywords:

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

Affiliations:
Gołubowska B.-IPPT PAN
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
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

Affiliations:
Gołubowska B.-IPPT PAN
Kovalchuk V.-IPPT PAN
Sławianowski J.J.-IPPT PAN
5.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, ISSN: 1531-3492, DOI: 10.3934/dcdsb.2012.17.699, Vol.17, No.2, pp.699-733, 2012
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

Affiliations:
Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
6.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Generalized Weyl–Wigner–Moyal–Ville formalism and topological groups, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.1531, Vol.35, pp.17-42, 2012
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:
Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
7.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)

Affiliations:
Sławianowski J.J.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
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: 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

Affiliations:
Gołubowska B.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
Sławianowski J.J.-IPPT PAN
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: 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

Affiliations:
Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
10.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Quasiclassical and Quantum Systems of Angular Momentum. Part 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

Affiliations:
Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
11.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Quasiclassical and Quantum Systems of Angular Momentum. Part 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

Affiliations:
Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
12.Sławianowski J.J., Kovalchuk V., 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

Affiliations:
Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
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: 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

Affiliations:
Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
14.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

Affiliations:
Sławianowski J.J.-IPPT PAN
Gołubowska B.-IPPT PAN
15.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

Affiliations:
Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Sławianowska A.-other affiliation
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
Zawistowski Z.J.-IPPT PAN
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 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

Affiliations:
Sławianowski J.J.-other affiliation
Kovalchuk V.-IPPT PAN
Sławianowska A.-other affiliation
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
Zawistowski Z.J.-IPPT PAN

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 non-Euclidean spaces, IPPT Reports on Fundamental Technological Research, 8, pp.1-129, 2006
2.
484
Sławianowski J.J., Kovalchuk V., Sławianowska A., Gołubowska B., Martens A., Rożko E.E., Zawistowski Z.J., Invariant geodetic systems on Lie groups and affine models of internal and collective degrees of freedom, Prace IPPT - IFTR Reports, Warszawa, 7, pp.1-164, 2004
List of chapters in recent monographs
1.
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
2.
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
3.
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
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.70-155, 2011

Conference papers
1.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

Affiliations:
Sławianowski J.J.-IPPT PAN
Gołubowska B.-IPPT PAN
2.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

Affiliations:
Gołubowska B.-IPPT PAN
Kovalchuk V.-IPPT PAN
Rożko E.E.-IPPT PAN
Sławianowski J.J.-IPPT PAN
3.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

Affiliations:
Sławianowski J.J.-IPPT PAN
Gołubowska B.-IPPT PAN