Invariant geodetic systems on Lie groups and affine models of internal and collective degrees of freedom

Sławianowski, J. J.; Kovalchuk, V.; Sławianowska, A.; Gołubowska, B.; Martens, A.; Rożko, E. E.; Zawistowski, Z. J.

Artykuł - szczegóły

Tytuł artykułu

Invariant geodetic systems on Lie groups and affine models of internal and collective degrees of freedom

Autorzy

Sławianowski J. J. , Kovalchuk V. , Sławianowska A. , Gołubowska B. , Martens A. , Rożko E. E. , Zawistowski Z. J.

Wybrane pełne teksty z tego czasopisma

http://prace.ippt.gov.pl/

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Warianty tytułu

Języki publikacji

Abstrakty

In some of our earlier papers including rather old ones we have discussed the concept of affinely-rigid body, i.e., continuous, discrete, or simply finite system of material points subject to such constraints that all affine relations between its elements are frozen during any admissible motion. For example, all material straight lines remain straight lines in the course of evolution, and their parallelism is also a constant, non-violated property. Unlike this, the metrical features, like distances and angles, need not be preserved. In other words, such a body is restricted in its behaviour to rigid translations, rigid rotations, and homogeneous deformations. Models of this kind may be successfully applied in a very wide spectrum of physical problems like nuclear dynamics (droplet model of the atomic nuclei), molecular vibrations, macroscopic elasticity (in situations when the length of excited waves is comparable with the size of the body), in the theory of microstructured bodies (micromorphic continua), in geophisics (the theory of the shape of Earth), and even in large-scale astropysics (vibrating stars, vibrating concentrations of the cosmic substratum, like galaxies or concentrations of the interstellar dust).

Słowa kluczowe

mechanika geometria stopnie swobody

Wydawca

Instytut Podstawowych Problemów Techniki PAN

Czasopismo

Prace Instytutu Podstawowych Problemów Techniki PAN

Rocznik

2004

Tom

nr 7

Strony

3--164

Opis fizyczny

Bibliogr. 91 poz.

Twórcy

autor

Sławianowski J. J.

Institute of Fundamental Technological Research PAS

autor

Kovalchuk V.

vkoval@ippt.pan.pl

Institute of Fundamental Technological Research PAS

autor

Sławianowska A.

Institute of Fundamental Technological Research PAS

autor

Gołubowska B.

Institute of Fundamental Technological Research PAS

autor

Martens A.

Institute of Fundamental Technological Research PAS

autor

Rożko E. E.

Institute of Fundamental Technological Research PAS

autor

Zawistowski Z. J.

Institute of Fundamental Technological Research PAS

Bibliografia

[1] R. Abraham and J. E. Marsden, Foundations of Mechanics (second ed.), The Benjamin-Cummings Publishing Company, London-Amsterdam-Sydney-Tokyo, 1978.
[2] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1978.
[3] D. Arsenović, A. O. Barut and M. Božić, The Critical Turning Points in the Solutions of the Magnetic-Top Equations of Motion, Il Nuovo Cimento 110B (1995), no. 2, 177-188.
[4| D. Arsenović, A. O. Barut, Z. Marić and M. Božić, Semi-Classical Quantization of the Magnetic Top, Il Nuovo Cimento 110B (1995), no. 2, 163-175.
[5] A. O. Barut, M. Božić and Z. Marić, The Magnetic Top as a Model of Quantum Spin, Annals of Physics 214 (1992), no. 1, 53-83.
[6] A. O. Barut and R. Rączka, Theory of Group Representations and Applications, PWN - Polish Scientific Publishers, Warsaw, 1977.
[7] E. Binz, Global Differential Geometric Methods in Elasticity and Hydrodynamics, in: Differential Geometry, Group Representations and Quantization, Lecture Notes in Physics, vol. 379, edited by J. D. Hennig, W. Lücke, and J. Tolar, Springer-Verlag, Berlin-Heidelberg, 1991.
[8] O. I. Bogoyavlensky, Methods of Qualitative Theory of Dynamical Systems in Astrophysics and Gas Dynamics, Springer, Berlin-Heidelberg-New York, 1985.
[9] A. Bohr and B. A. Mottelson, Nuclear Structure. Volume II, W. A. Benjamin, Reading, Mass., 1975.
[10] F. Calogero and C. J. Marchioro, Ezact Solution of a One-Dimensional Three-Body Scattering Problem with Two-Body and/or Three-Body Inverse-Square Potentials, Math. Phys. 15 (1974), 1425.
[11] G. Capriz, Continua with Microstructure, Springer Tracts in Natural Philosophy, vol. 35, Springer-Verlag, New York-Berlin-Heidelberg-Paris-lokyo, 1989.
[12] G. Capriz, Continua with Substructure, Phys. Mesomech. 3 (2000), 5-14, 37-50.
[13] G. Capriz and P. M. Mariano, Balance at a Junction among Coherent Interfaces in Materials with Substructure, in: G. Capriz and P. M. Mariano (eds), Advances in Multifield Theories of Materials with Substructure, Birkhauser, Basel, 2003.
[14] G. Capriz and P. M. Mariano, Symmetries and Hamiltonian Formalism for Complex Materials, Journal of Elasticity 72 (2003), 57-70.
[15] S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Yale Univ. Press, New Haven-London, 1969.
[16] F. J. Dyson, Dynamics of a Spinning Gas Cloud, Journal of Mathematics and Mechanics 18 (1968), no. 1, 91.
[17] D. G. Ebin, The Motion of Slightly Compressible Fluids Viewed as a Motion with Strong Constraining Force, Ann. Math. 105 (1977), 141-200.
[18] D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. 92 (1970), 102-163.
[19] A. C. Eringen, Nonlinear Theory of Continuous Media, McGraw-Hill Book Company, New York, 1962.
[20] A. C. Eringen, Mechanics of Micromorphic Continua, in: Proceedings of the IUTAM Symposium on Mechanics of Generalized Continua, Freudenstadt and Stuttgart, 1967, editor: E. Kröner, vol. 18, Springer, Berlin-Heidelberg-New York, 1968, 18-38.
[21] K. Frąckiewicz, M. Seredyńska and J. J. Sławianowski, Controlled Motion of Finite Elements, Journal of Theoretical and Applied Mechanics 29 (1991), no. 3-4, 671-686.
[22] P. Godlewski, Quantization of Anisotropic Rigid Body, Int. J. Theor. Phys. 42 (2003), no. 12, 2863-2875.
[23| H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass, 1950.
[24] B. Golubowska, Motion of Test Rigid Bodies in Riemannian Spaces, Reports on Mathematical Physics 48 (2001), no. 1/2, 95-102.
[25] B. Golubowska, Models of Internal Degrees of Freedom Based on Classical Groups and Their Homogeneous Spaces, Reports on Mathematical Physics 49 (2002), no. 2/3, 193-201.
[26] B. Golubowska, Affine Models of Internal Degrees of Freedom and Their Action-Angle Description, Reports on Mathematical Physics 51 (2003), no. 2/3, 205-214.
[27] B. Golubowska, Infinitesimal Affinely-Rigid Bodies in Riemann Spaces, in: Proceedings of Institute of Mathematics of NAS of Ukraine, editors: A. G. Nikitin, V. M. Boyko, R. O. Popovych, and I. A. Yehorchenko, vol. 50, no. 2, Institute of Mathematics, Kyiv, 2004, 774-779.
[28] B. Golubowska, Action-Angle Analysis of Some Geometric Models of Internal Degrees of Freedom, Journal of Nonlinear Mathematical Physics 11 (2004), Supplement, 138-144.
[29] R. Gutowski, Analytical Mechanics, PWN - Polish Scientific Publishers, Warsaw, 1971, (in Polish).
[30] F. W. Hehl, G. D. Kerlick and P. Van der Heyde, Physical Review D 10 (1974), 1066.
[31] F. W. Hehl, E. A. Lord and Y. Ne'eman, Hadron Dilatation, Shear and Spin as Components of the Intrinsic Hypermomentum. Current and Metric-Affine Theory of Gravitation, Physics Letters 71B (1977), 432.
[32] R. Hermann, Differential Geometry and the Calculus of Variations, Academic Press, New York, 1968.
[33] R. Hermann, Vector Bundles in Mathematical Physics, W. A. Benjamin, New York, 1970.
[34| S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, New York, 1963.
[35] E. Kossecka and H. Zorski, Int. J. Solids Struct. 3 (1967), 881.
[36] L. D. Landau and E. M. Lifshitz, Continuous Media Mechanics, PWN - Polish Scientific Publishers, Warsaw, 1958, (in Polish).
[37] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon Press, London, 1958.
[38] L. H. Loomis, An Introduction to Abstract Harmonic Analysis, D. Van Nostrand Company, Inc., Princeton-New Jersey-Toronto-London-New York, 1953.
[39] G. W. Mackey, The Mathematical Foundations of Quantum Mechanics, Benjamin, New York, 1963.
[40| P. M. Mariano, Multifield Theories in Mechanics of Solids, Adv. Appl. Mech. 38 (2001), 1-93.
[41] P. M. Mariano, Cancellation of Vorticity in Steady-State Non-lso-entropic Flows of Complex Fluids, J. Phys. A: Math. Gen. 36 (2003), 9961-9972.
[42] J. E. Marsden, Lectures on Geometric Methods in Mathematical Physics, SIAM, Philadelphia, 1981.
[43] J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Clifis, New York, 1983.
[44] J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Springer, New York, 1994.
[45] J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems (second ed.), Springer, New York, 1999.
[46] A. Martens, Dynamics of Holonomically Constrained Affinely-Rigid Body, Reports on Mathematical Physics 49 (2002), no. 2/3, 295-303.
[47] A. Martens, Quantization of Affinely-Rigid Body with Constraints, Reports on Mathematical Physics 51 (2003), no. 2/3, 287-295.
[48] A. Martens, Hamiltonian Dynamics of Planar Affinely-Rigid Body, Journal of Nonlinear Mathematical Physics 11 (2004), Supplement, 145-150.
[49] A. Martens, Quantization of the Planar Affinely-Rigid Body, Journal of Nonlinear Mathematical Physics 11 (2004), Supplement, 151-156.
[50| K. Maurin, General Eigenfunction Ezpansions and Unitary Representations of Topological Groups, PWN - Polish Scientific Publishers, Warsaw, 1968.
[51] J. Moser, Dynamical systems theory and applications, Lecture Notes in Physics, vol. 38, Springer, Berlin, 1975.
[52] J. Moser, Three Integrable Hamiltonian Systems Connected with lsospectral Deformations, Advances in Math. 16 (1975), 197-220.
[53] T. Ratiu, Am. J. Math. 104 (1982), 409-448.
[54] M. Roberts, C. Wulff and J. Lamb, Hamiltonian Systems Near Relative Equilibria, Journal of Differential Equations 179 (2002), 562-604.
[55] M. E. Rose, Elementary Theory of Angular Momentum, Dover Publications, 1995.
[56] A. K. Sławianowska, On Certain Nonlinear Many-Body Problems on Lines and Circles, Arch. of Mech. 41 (1989), no. 5, 619-640.
[57] A. K. Sławianowska and J. J. Sławianowski, Quantization of Affinely Rigid Body in N Dimensions, Reports on Mathematical Physics 29 (1991), no. 3, 297-320.
[58] J. J. Sławianowski, Analytical Mechanics of Homogeneous Deformations, Prace IPPT - IFTR Reports, no. 8, Warsaw, 1973, (in Polish).
[59] J. J. Sławianowski, Classical Pure States: Information and Symmetry in Statistical Mechanics, International Journal of Theoretical Physics 8 (1973), no. 6, 451-462.
[60] J. J. Sławianowski, Analytical Mechanics of Finite Homogeneous Strains, Archives of Mechanics 26 (1974), no. 4, 569-587.
[61] J. J. Sławianowski, Homogeneously Deformable Body in a Curved Space, Bulletin de I’Académie Polonaise des Sciences, Série des sciences techniques 23 (1975), no. 2, 43-47.
[62] J. J. Sławianowski, The Mechanics of an Affinely-Rigid Body, International Journal of Theoretical Physics 12 (1975), no. 4, 271-296.
[63] J. J. Sławianowski, Newtonian Dynamics of Homogeneous Strains, Archives of Mechanics 27 (1975), no. 1, 93-102.
[64] J. J. Sławianowski, Newtonian Dynamics of Polynomial Deformations, Bulletin de I’Académie Polonaise des Sciences, Série des sciences techniques 23 (1975), no. 1, 17-22.
[65] J. J. Sławianowski, Analytical Mechanics of Deformable Bodies, PWN - Polish Scientific Publishers, Warszawa-Poznań, 1982, (in Polish).
[66] J. J. Sławianowski, The Mechanics of the Homogeneously-Deformable Body. Dynamical Models with High Symmetries, Zeitschrift für Angewandte Mathematik und Mechanik 62 (1982), 229-240.
[67] J. J. Sławianowski, Lie-Algebraic Solutions of Affinely-Invariant Equations for the Field of Linear Frames, Reports on Mathematical Physics 23 (1986), no. 2, 177-197.
[68] J. J. Sławianowski, Affinely-Rigid Body and Hamiltonian Systems on GL(n, R), Reports on Mathematical Physics 26 (1988), no. 1, 73-119.
[69] J. J. Sławianowski, Geometry of Phase Spaces, John Wiley & Sons, Chichester-New York-Brisbane-Toronto-Singapore, 1991.
[70] J. J. Sławianowski, Nonlinear Vibrations of Rigid Bodies. Projective Correspondence between Rigid Body and Material Point Mechanics, in: Proceedings of the 2nd Polish-German Workshop on Dynamical Problems in Mechanical Systems, March 10-17, 1991, in Paderborn, editors: R. Bogacz, J. Lücker, K. Popp, IFTR-Editions, Warsaw, 1991, 25-34.
[71] J. J. Sławianowski, One-Dimensional Chains, Matria Groups and Quantization of Poisson Structures, Journal of Technical Physics 39 (1998), no. 2, 163-187.
[72] J. J. Sławianowski, Group-Theoretic Approach to Internal and Collective Degrees of Freedom in Mechanics and Field Theory, Technische Mechanik 22 (2002), no. 1, 8-13.
[73] J. J. Sławianowski, Quantum and Classical Models Based on GL(n,R)-Symmetry, in: Proceedings of the Second International Symposium on Quantum Theory and Symmetries, Kraków, Poland, July 18-21, 2001, editors: E. Kapuścik and A. Horzela, World Scientific, New Jersey-London-Singapore-Hong Kong, 2002, 582-588.
[74] J. J. Sławianowski, Linear Frames in Manifolds, Riemannian Structures and Description of Internal Degrees of Freedom, Reports on Mathematical Physics 51 (2003), no. 2/3, 345-369.
[75] J. J. Sławianowski, Classical and Quantum Collective Dynamics of Deformable Objects. Symmetry and Integrability Problems, in: Proceedings of the Fifth International Conference on Geometry, Integrability and Quantization, June 5-12, 2003, Varna, Bulgaria, editors: Ivaïlo M. Mladenov and Allen C. Hirshfeld, SOFTEX, Sofia, 2004, 81-108.
[76] J. J. Sławianowski, Geodetic Systems on Linear and Affine Groups. Classics and Quantization, Journal of Nonlinear Mathematical Physics 11 (2004), Supplement, 130-137.
[77] J. J. Sławianowski and V. Kovalchuk, Invariant Geodetic Problems on the Affine Group and Related Hamiltonian Systems, Reports on Mathematical Physics 51 (2003), no. 2/3, 371-379.
[78] J. J. Sławianowski and V. Kovalchuk, Invariant Geodetic Problems on the Projective Group Pr(n, R), in: Proceedings of Institute of Mathematics of NAS of Ukraine, editors: A. G. Nikitin, V. M. Boyko, R. O. Popovych, and I. A. Yehorchenko, vol. 50, no. 2, Institute of Mathematics, Kyiv, 2004, 955-960.
[79] J. J. Sławianowski and V. Kovalchuk, Classical and Quantized Affine Physics: A Step Towards It, Journal of Nonlinear Mathematical Physics 11 (2004), Supplement, 157-166.
[80] J. J. Sławianowski and A. K. Sławianowska, Virial Coefficients, Collective Models and Problems with the Galerkin Procedure, Archives of Mechanics 45 (1993), no. 3, 305-331.
[81| J. J. Sławianowski and A. K. Sławianowska, Hamiltonian Systems on Linear Groups and One-Dimensional Lattices with Internal Parameters, Machine Dynamics Problems 20 (1998), 263-273.
[82] J. L. Synge, Classical Dynamics, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960.
[83| M. Toda, Theory of Nonlinear Lattices, Springer-Verlag, Berlin-Heidelberg-New York, 1981.
[84| C. Trimarco, Microscopic Variables and Macroscopic Quantities, in: Proceedings of Workshop on Greometry, Continua and Microstructures, Paris, May 28-29, Hermann, Paris, 1997.
[85| K. Westpfahl, Annalen der Physik 20 (1967), 113.
[86] E. P. Wigner, Gruppentheorie und Ihre Anwendung auf die Quantenmechanik der Atomspektren, F. Viewag und Sohn, Braunschweig, 1931, (eng. translation by J. J. Griffin, Academic Press, New York, 1959).
[87] E. P. Wigner, in: Quantum Theory of Angular Momentum, editors: L. C. Biedenharn and H. van Dam, Academic Press, New York, 1965.
[88] W. Wojtyński, Lie Groups and Algebras, PWN - Polish Scientific Publishers, Warsaw, 1986, (in Polish).
[89] C. Wulff and M. Roberts, Hamiltonian Systems Near Relative Periodic Orbits, SLAM Journal of Dynamical Systems 1 (2002), no. 1, 1-43.
[90] D. P. Zhelobenko, Compact Lie Groups and Their Representations, Translations of Mathematical Monographs, vol. 40, AMS, 1978.
[91] D. P. Zhelobenko, Representations of Lie Groups, Nauka, Moscow, 1983, (in Russian).

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Bibliografia

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