1. |
Pulov V.^{♦}, Kowalczuk W., Mladenov I.M.^{♦}, Geometry of Enumerable Class of Surfaces Associated with Mylar Balloons,
Mathematics, ISSN: 2227-7390, DOI: 10.3390/math12040557, Vol.12, No.4, pp.557-1-18, 2024Abstract: In this paper, the very fundamental geometrical characteristics of the Mylar balloon like the profile curve, height, volume, arclength, surface area, crimping factor, etc. are recognized as geometrical moments ℐ Keywords: Mylar balloons, geometrical moments, elliptic integrals, beta and gamma functions, recursive relations, crimping factor, lemniscate constant Affiliations:
Pulov V. | - | other affiliation | Kowalczuk W. | - | IPPT PAN | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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2. |
Kowalczuk W., Mladenov I.M.^{♦}, Comparison of main geometric characteristics of deformed sphere and standard spheroid,
BULLETIN OF THE POLISH ACADEMY OF SCIENCES: TECHNICAL SCIENCES, ISSN: 0239-7528, DOI: 10.24425/bpasts.2023.147058, pp.1-8, 2023Abstract: Here we compare the geometric descriptions of the deformed sphere (i.e., the so-called λ-sphere) and the standard spheroid (namely, World Geodetic System 1984’s reference ellipsoid of revolution). Among the main geometric characteristics of those two surfaces of revolution embedded into the three-dimensional Euclidean space we consider the semi-major (equatorial) and semi-minor (polar) axes, quartermeridian length, surface area, volume, sphericity index, and tipping (bifurcation) point for geodesics. Next, the RMS (Root Mean Square) error is defined as the square-rooted arithmetic mean of the squared relative errors for the individual pairs of the discussed six main geometric characteristics. As a result of the process of minimization of the RMS error, we have obtained the proposition of the optimized value of the deformation parameter of the λ-sphere, for which we have calculated the absolute and relative errors for the individual pairs of the discussed main geometric characteristics of λ-sphere and standard spheroid (the relative errors are of the order of 10−6 – 10−9). Among others, it turns out that the value of the flattening factor of the spheroid is quite a good approximation for the corresponding value of the deformation parameter of the λ-sphere (the relative error is of the order of 10−4). Keywords: deformed sphere, standard spheroid, sphericity index, tipping (bifurcation) point for geodesics, elliptic integrals and functions Affiliations:
Kowalczuk W. | - | IPPT PAN | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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3. |
Kowalczuk W., Pulov V.^{♦}, Mladenov I.M.^{♦}, Mylar Balloon and Associated Geometro-Mechanical Moments,
Mathematics, ISSN: 2227-7390, DOI: 10.3390/math11122646, Vol.11, No.12, pp.2646-1-13, 2023Abstract: Starting with identifications of the very fundamental geometric characteristics of a Mylar balloon such as the profile curve, height, volume, arclength, surface area, crimping factor, etc., using the geometrical moments In(x) and In, we present explicit formulas for them and those of the mechanical moments of both solid and hollow balloons of arbitrary order. This is achieved by relying on the recursive relationships among elliptic integrals and the final results are expressed via the fundamental mathematical constants such as π, lemniscate constant ˜ω, and Gauss’s constant G. An interesting periodicity modulo 4 was detected and accounted for in the final formulas for the moments. The principal results are illustrated by two tables, a few graphics, and some direct relationships with other fundamental areas in mathematics, physics and geometry are pointed out. Keywords: solid and hollow Mylar balloons, crimping factor, geometro-mechanical moments, recursive relations, elliptic integrals and functions, gamma functions, Gauss’s and lemniscate constants Affiliations:
Kowalczuk W. | - | IPPT PAN | Pulov V. | - | other affiliation | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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4. |
Kowalczuk W., Mladenov I.M.^{♦}, Explicit solutions for geodetic problems on the deformed sphere as reference model for the geoid,
Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-25-2023-73-94, Vol.25, pp.1-23, 2023Abstract: In this article, we consider deformed spheres as a new reference model for the geoid, alternatively to the classical ellipsoidal one. The parametrization of deformed spheres is furnished through the incomplete elliptic integrals. From the other side, the solutions for geodesics on those surfaces are given entirely via elementary analytical functions, contrary to the case of ellipsoids of revolution. We explicitly described algorithms (all necessary computational steps) for the solution of the direct and inverse geodetic problems on the deformed spheres. Finally, we presented a few illustrative numerical solutions of the inverse geodetic problems for two conceptual cases of near and far points. It had turned out that even in the non-optimized case we obtained the good agreement with the predictions of the World Geodetic System 1984’s ellipsoidal reference model. Keywords: deformed spheres, direct and inverse geodetic problems, geodesy and navigation problems, reference models for geoid, World Geodetic System 1984 Affiliations:
Kowalczuk W. | - | IPPT PAN | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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5. |
Kovalchuk V., Mladenov I.M.^{♦}, λ-spheres as a new reference model for the geoid,
MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.8782, pp.1-14, 2022Abstract: We propose here a new reference model approximating the actual shape of the Earth. It is based on Keywords: closed and open geodesics, deformed spheres, geodesy and mapping problems, geoid's reference models, geometric flattening parameter, incomplete elliptic integrals Affiliations:
Kovalchuk V. | - | IPPT PAN | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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6. |
Kovalchuk V., Chudziński P., Unifying approach to Heisenberg models for spin distributions on curved surfaces of revolution with isothermal (conformal) metrics,
MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.8587, pp.1-13, 2022Abstract: In the present article, a unifying approach to description of different Heisenberg models (i.e., XY model, isotropic case, easy-plane, and easy-axis regimes) for spin distributions on curved surfaces is discussed. After introduction of isothermal (conformal) metrics on surfaces of revolution, the axially symmetric solution of the Euler–Lagrange equation for the quasi-one-dimensional magnetic Hamiltonian has been obtained for all four discussed cases in the form of an incomplete elliptic integral of the third kind. It has been shown that the limiting case of XY model can be achieved from an easy-plane regime when the parameter of anisotropy of interaction amongst spins ? is approaching −1, while the isotropic case can be achieved from both easy-plane and easy-axis regimes when ? is approaching 0. After the application of the corresponding boundary conditions, the soliton-like solutions have been obtained that either cover the whole surface (in a one-twist or multi-twist manner) or can be arranged in the form of a periodic lattice structure on the curved surface. The general unifying scheme has been illustrated using the exemplary surfaces of revolution with zero (cylinder), negative (catenoid), and positive (sphere) Gaussian curvatures. Keywords: elliptic integrals and elliptic functions, Heisenberg models for spin distributions, isothermal (conformal) metrics, soliton-like solutions, surfaces of revolution Affiliations:
Kovalchuk V. | - | IPPT PAN | Chudziński P. | - | IPPT PAN |
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7. |
Kovalchuk V., Mladenov I.M.^{♦}, Mechanics of incompressible test bodies moving on λ-spheres,
MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.8126, pp.1-14, 2022Abstract: The considerations of the mechanics of incompressible test bodies moving on spheres are generalized to the case of λ-spheres. Both surfaces are examples of Riemannian manifolds realized as two-dimensional manifolds with strictly positive Gaussian curvature under the condition λ < 1/3 for the λ-spheres. A convenient parametrization of λ-spheres embedded as two-dimensional surfaces of revolution into the three-dimensional Euclidean space is derived. The so-obtained parametrization is expressed in a concise form via the elliptic integrals of the first, second, and third kind. Next, the geodetic motion is considered. The explicit solutions for two branches of the incompressible motion are obtained in the parametric form. For the special case of geodetics corresponding to meridians taken as geodesics their analysis is performed in detail. In the latter case, the so-obtained geodetic solutions are reduced to the incomplete elliptic integrals of the first, second, and third kind. Keywords: λ-sphere as a generalization of the usual sphere, elliptic integrals and elliptic functions, geodesic and geodetic equations of motion, incompressibility constraints, mechanics of infinitesimal test bodies Affiliations:
Kovalchuk V. | - | IPPT PAN | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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8. |
Kovalchuk V., Mladenov I.M.^{♦}, λ-spheres as a new reference model for geoid: explicit solutions of the direct and inverse problems for loxodromes (rhumb lines),
Mathematics, ISSN: 2227-7390, DOI: 10.3390/math10183356, Vol.10, No.18, pp.3356-1-3356-10, 2022Abstract: In this paper, we present a new reference model that approximates the actual shape of the Earth, based on the concept of the deformed spheres with the deformation parameter λ. These surfaces, which are called λ-spheres, were introduced in another setting by Faridi and Schucking as an alternative to the spheroids (i.e., ellipsoids of revolution). Using their explicit parametrizations that we have derived in our previous papers, here we have defined the corresponding isothermal (conformal) coordinates as well as obtained and solved the differential equation describing the loxodromes (or rhumb lines) on such surfaces. Next, the direct and inverse problems for loxodromes have been formulated and the explicit solutions for azimuths and arc lengths have been presented. Using these explicit solutions, we have assessed the value of the deformation parameter λ for our reference model on the basis of the values for the semi-major axis of the Earth a and the quarter-meridian mp (i.e., the distance between the Equator and the North or South Pole) for the current best ellipsoidal reference model for the geoid, i.e., WGS 84 (World Geodetic System 1984). The latter is designed for use as the reference system for the GPS (Global Positioning System). Finally, we have compared the results obtained with the use of the newly proposed reference model for the geoid with the corresponding results for the ellipsoidal (WGS 84) and spherical reference models used in the literature. Keywords: deformed spheres, incomplete elliptic integrals, geoid’s reference models, loxodromes or rhumb lines, azimuths and arc lengths, geodesy and navigation problems Affiliations:
Kovalchuk V. | - | IPPT PAN | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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9. |
Kovalchuk V., Gołubowska B., Mladenov I.M.^{♦}, Mechanics of infinitesimal gyroscopes on helicoid-catenoid deformation family of minimal surfaces,
BULLETIN OF THE POLISH ACADEMY OF SCIENCES: TECHNICAL SCIENCES, ISSN: 0239-7528, DOI: 10.24425/bpasts.2021.136727, Vol.69, No.2, pp.e136727-1-9, 2021Abstract: In this paper we explore the mechanics of infinitesimal gyroscopes (test bodies with internal degrees of freedom) moving on an arbitrary member of the helicoid-catenoid family of minimal surfaces. As the configurational spaces within this family are far from being trivial manifolds, the problem of finding the geodesic and geodetic motions presents a real challenge. We have succeeded in finding the solutions of those motions in an explicit parametric form. It is shown that in both cases the solutions can be expressed through the elliptic functions and elliptic integrals, but in the geodetic case some appropriately chosen compatibility conditions for glueing together different branches of the solution are needed. Additionally, an action-angle analysis of the corresponding Hamilton-Jacobi equations is performed for external potentials that are well-suited to the geometry of the problem under consideration. As a result, five different sets of conditions between the three action variables and the total energy of the infinitesimal gyroscopes are obtained. Keywords: action-angle analysis, mechanics of infinitesimal gyroscopes, geodesic and geodetic equations of motion, helicoid-catenoid deformation family of minimal surfaces, elliptic integrals and elliptic functions Affiliations:
Kovalchuk V. | - | IPPT PAN | Gołubowska B. | - | IPPT PAN | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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10. |
Kovalchuk V., Gołubowska B., Rożko E.E., Mechanics of incompressible test bodies moving in Riemannian spaces,
MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.6651, pp.1-15, 2020Abstract: In the present paper, we have discussed the mechanics of incompressible test bodies moving in Riemannian spaces with nontrivial curvature tensors. For Hamilton's equations of motion, the solutions have been obtained in the parametrical form and the special case of the purely gyroscopic motion on the sphere has been discussed. For the geodetic case when the potential is equal to zero, the comparison between the geodetic and geodesic solutions has been done and illustrated in details for the case of a particular choice of the constants of motion of the problem. The obtained results could be applied, among others, in geophysical problems, for example, for description of the movement of continental plates or the motion of a drop of fat or a spot of oil on the surface of the ocean (e.g., produced during some "ecological disaster"), or generally in biomechanical problems, for example, for description of the motion of objects with internal structure on different curved two-dimensional surfaces (e.g., transport of proteins along the curved biological membranes). Keywords: geodesics, geodetics, gyroscopic motion, incompressibility constraints, mechanics of infinitesimal test bodies, Riemannian spaces Affiliations:
Kovalchuk V. | - | IPPT PAN | Gołubowska B. | - | IPPT PAN | Rożko E.E. | - | IPPT PAN |
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11. |
Kovalchuk V., Mladenov I.M.^{♦}, Classical motions of infinitesimal rotators on mylar balloons,
MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.6660, pp.1-14, 2020Abstract: This paper starts with the derivation of the most general equations of motion for the infinitesimal rotators moving on arbitrary two‐dimensional surfaces of revolution. Both geodesic and geodetic (i.e., without any external potential) equations of motion on surfaces with nontrivial curvatures that are embedded into the three dimensional Euclidean space are discussed. The Mylar balloon as a concrete example for the application of the scheme was chosen. A new parameterization of this surface is presented, and the corresponding equations of motion for geodesics and geodetics are expressed in an analytical form through the elliptic functions and elliptic integrals. The so‐obtained results are also compared with those for the two‐dimensional sphere embedded into the three‐dimensional Euclidean space for which it can be shown that the geodesics and geodetics are plane curves realized as the great and small circles on the sphere, respectively. Keywords: elliptic integrals and elliptic functions, geodesics and geodetics, infinitesimal rotators, mylar balloons, non-Euclidean spaces, surfaces of revolution Affiliations:
Kovalchuk V. | - | IPPT PAN | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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12. |
Kovalchuk V., Mladenov I.M.^{♦}, Mechanics of infinitesimal gyroscopes on mylar balloons and their action-angle analysis,
MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.6099, Vol.43, No.6, pp.3040-3051, 2020Abstract: Here, we apply the general scheme for description of mechanics of infinitesimal bodies in Riemannian spaces to the example of geodetic and non-geodetic (for two different model potentials) motions of infinitesimal rotators on the Mylar balloon. The structure of partial degeneracy is investigated with the help of the corresponding Hamilton-Jacobi equation and action-angle analysis. In all situations it was found that for any of the sixth disjoint regions in the phase space among the three action variables only two are essential for the description of our models at the level of the old quantum theory (according to the Bohr-Sommerfeld postulates). Moreover, in both non-geodetic models, the action variables were intertwined with the quantum number N corresponding to the quantized value of the radius r of the inflated Mylar balloon. Keywords: action-angle analysis, affinely-rigid bodies, Bohr-Sommerfeld quantum theory, Hamilton-Jacobi equation, infinitesimal gyroscopes, mylar balloons, residue analysis Affiliations:
Kovalchuk V. | - | IPPT PAN | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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13. |
Kovalchuk V., Gołubowska B., Mladenov I.M.^{♦}, Mechanics of infinitesimal test bodies on Delaunay surfaces: spheres and cylinders as limits of unduloids and their action-angle analysis,
Journal of Geometry and Symmetry in Physics, ISSN: 1312-5192, DOI: 10.7546/jgsp-53-2019-55-84, Vol.53, pp.55-84, 2019Abstract: This paper discusses the motion of infinitesimal gyroscopes along the two-dimensional surfaces of constant mean curvature embedded into the three-dimensional Euclidean space. We have considered the cases of unduloids, spheres, and cylinders for which the corresponding Hamilton-Jacobi equations are written and analyzed with the help of the action-angle variables. Spheres and cylinders are considered as limiting cases of unduloids and the residue analysis was performed which provides the connection between all three action variables and the energy. This has been traced also for the geodetic situations and for two additional classical model potentials. Keywords: action-angle variables, affinely-rigid bodies, d'Alembert mechanics, Delaunay surfaces, geometry of curves and surfaces, Hamilton-Jacobi equation, harmonic/anharmonic oscillator potential models, infinitesimal test bodies, residue analysis, Riemannian spaces Affiliations:
Kovalchuk V. | - | IPPT PAN | Gołubowska B. | - | IPPT PAN | Mladenov I.M. | - | Institute of Mechanics, Bulgarian Academy of Science (BG) |
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14. |
Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Space‐time as a structured relativistic continuum,
MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.5087, Vol.41, pp.5404-5422, 2018Abstract: There are various models of gravitation: the metrical Hilbert‐Einstein theory, a wide class of intrinsically Lorentz‐invariant tetrad theories (generally covariant in the space‐time sense), and many gauge models based on various internal symmetry groups (Lorentz, Poincare, GL(n,R), SU(2,2), GL(4,C), etc). The gauge models are usually preferred but nevertheless it is an interesting idea to develop the class of GL(4,R)‐invariant (or rather GL(n,R)‐invariant) tetrad (n‐leg) generally covariant models. This is done below and motivated by our idea of bringing back to life the Thales of Miletus concept of affine symmetry. Formally, the obtained scheme is a generally covariant tetrad (n‐leg) model, but it turns out that generally covariant and intrinsically affinely invariant models must have a kind of nonaccidental Born‐Infeld‐like structure. Let us also mention that they, being based on tetrads (n‐legs), have many features common with continuous defect theories. It is interesting that they possess some group‐theoretical solutions and more general spherically symmetric solutions, discussion of which is the main new result presented in this paper, including the applications of the 't Hooft‐Polyakov monopoles in the generally covariant theories, which enables us to find some rigorous solutions of our strongly nonlinear equations. It is also interesting that within such a framework, the normal‐hyperbolic signature of the space‐time metric is not introduced by hand but appears as a kind of solution, rather integration constants, of differential equations. Let us mention that our Born‐Infeld scheme is more general than alternative tetrad models. It may be also used within more general schemes, including also the gauge ones. Keywords: micromorphic medium, modified gravity, relativistic continuum, spherically symmetric solutions, theory of fundamental interactions. Affiliations:
Sławianowski J.J. | - | IPPT PAN | Kovalchuk V. | - | IPPT PAN | Gołubowska B. | - | IPPT PAN | Martens A. | - | IPPT PAN | Rożko E.E. | - | IPPT PAN |
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15. |
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, No.1, pp.493-520, 2017Abstract: 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 |
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16. |
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, Vol.40, No.18, pp.6900-6918, 2017Abstract: 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 |
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17. |
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, 2014Abstract: 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 |
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18. |
Popov A.^{♦}, Kovalchuk V., Parametric representation of wave propagation in non-uniform media (both in transmission and stop bands),
MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.2687, Vol.36, No.11, pp.1350-1362, 2013Abstract: An analytical approach based on the parametric representation of the wave propagation in non-uniform media was considered. In addition to the previously developed theory of parametric antiresonance describing the field attenuation in stop bands, in the present paper, the behaviour of the Bloch wave in a transmission band was investigated. A wide class of exact solutions was found, and the correspondence to the quasi-periodic Floquet solutions was shown. Keywords: Floquet theorem, parametric resonance/antiresonance, wave propagation, non-uniform media, periodic structures, stop and transmission bands Affiliations:
Popov A. | - | Pushkov Institute of Terrestrial Magnetism, Ionosphere and RadioWave Propagation, Russian Academy of Sciences (RU) | Kovalchuk V. | - | IPPT PAN |
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19. |
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, 2012Abstract: 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 |
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20. |
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, 2012Abstract: 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 |
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21. |
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, 2012Abstract: 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 |
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22. |
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, 2011Abstract: 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 |
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23. |
Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Quasiclassical and Quantum Systems of Angular Momentum. Part III. Group Algebra su(2), Quantum Angular Momentum and Quasiclassical Asymptotics,
Journal of Geometry and Symmetry, ISSN: 1312-5192, DOI: 10.7546/jgsp-23-2011-59-95, Vol.23, pp.59-95, 2011Abstract: 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 |
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24. |
Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Quasiclassical and Quantum Systems of Angular Momentum. Part II. Quantum Mechanics on Lie Groups and Methods of Group Algebras,
Journal of Geometry and Symmetry, ISSN: 1312-5192, DOI: 10.7546/jgsp-22-2011-67-94, Vol.22, pp.67-94, 2011Abstract: 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 |
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25. |
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, ISSN: 1312-5192, DOI: 10.7546/jgsp-21-2011-61-94, Vol.21, pp.61-94, 2011Abstract: 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 |
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26. |
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, 2010Abstract: 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 |
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27. |
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, 2010Abstract: 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 Affiliations:
Sławianowski J.J. | - | IPPT PAN | Kovalchuk V. | - | IPPT PAN |
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28. |
Kovalchuk V., On classical dynamics of affinely-rigid bodies subject to the Kirchhoff-Love constraints,
Symmetry, Integrability and Geometry: Methods and Applications SIGMA, ISSN: 1815-0659, DOI: 10.3842/SIGMA.2010.031, Vol.6, No.031, pp.1-12, 2010Abstract: In this article we consider the affinely-rigid body moving in the three-dimensional physical space and subject to the Kirchhoff–Love constraints, i.e., while it deforms homogeneously in the two-dimensional central plane of the body it simultaneously performs one-dimensional oscillations orthogonal to this central plane. For the polar decomposition we obtain the stationary ellipsoids as special solutions of the general, strongly nonlinear equations of motion. It is also shown that these solutions are conceptually different from those obtained earlier for the two-polar (singular value) decomposition. Keywords: Affinely-rigid bodies with degenerate dimension, Kirchhoff–Love constraints, polar decomposition, Green deformation tensor, deformation invariants, stationary ellipsoids as special solutions Affiliations:
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29. |
Kovalchuk V., Rożko E.E., Classical models of affinely-rigid bodies with "thickness" in degenerate dimension,
Journal of Geometry and Symmetry in Physics, ISSN: 1312-5192, DOI: 10.7546/jgsp-14-2009-51-65, Vol.14, pp.51-65, 2009Abstract: The special interest is devoted to such situations when the material space of object with affine degrees of freedom has generally lower dimension than the one of the physical space. In other words when we have m-dimensional affinely-rigid body moving in the n-dimensional physical space, m < n. We mainly concentrate on the physical situation m = 2, n = 3 when “thickness” of flat bodies performs one-dimensional oscillations orthogonal to the two-dimensional central plane of the body. For the isotropic case in two “flat” dimensions some special solutions, namely, the stationary ellipses, which are analogous to the ellipsoidal figures of equilibrium well known in astro- and geophysics, e.g., in the theory of the Earth’s shape, are obtained. Keywords: Affinely-rigid bodies with degenerate dimension, non-zeroth thickness, two-polar decomposition, stationary ellipsoidal figures of equilibrium Affiliations:
Kovalchuk V. | - | IPPT PAN | Rożko E.E. | - | IPPT PAN |
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30. |
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, 2008Abstract: 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 Affiliations:
Kovalchuk V. | - | IPPT PAN | Sławianowski J.J. | - | IPPT PAN |
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31. |
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, 2005Abstract: 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 |
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32. |
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, 2004Abstract: 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 |
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33. |
Kovalchuk V., Green function for Klein-Gordon-Dirac equation,
Journal of Nonlinear Mathematical Physics, ISSN: 1402-9251, DOI: 10.2991/jnmp.2004.11.s1.9, Vol.11, No.Supplement, pp.72-77, 2004Abstract: The Green function for Klein-Gordon-Dirac equation is obtained. The case with the dominating Klein-Gordon term is considered. There seems to be a formal analogy between our problem and a certain problem for a 4-dimensional particle moving in the external field. The explicit relations between the wave function, Green function and initial conditions are established with the help of the T-exponent formalism. Keywords: Green functions, Klein-Gordon-Dirac equation, T-exponent formalism, Dirac plane harmonic waves Affiliations:
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34. |
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, 2004Abstract: 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 Affiliations:
Sławianowski J.J. | - | IPPT PAN | Kovalchuk V. | - | IPPT PAN |
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35. |
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, 2003Abstract: 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 Affiliations:
Sławianowski J.J. | - | IPPT PAN | Kovalchuk V. | - | IPPT PAN |
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36. |
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, 2002Abstract: 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 Affiliations:
Sławianowski J.J. | - | IPPT PAN | Kovalchuk V. | - | IPPT PAN |
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