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The co-divergence of vector valued currents
Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics
1. | Institute of Fundamental Technological Research, Polish Academy of Sciences, 5B, Pawińskiego str., 02-106 Warsaw, Poland, Poland, Poland, Poland, Poland |
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, revised and enlarged, With the assistance of Tudor Raţiu and Richard Cushman, Benjamin-Cummings Publishing Company, Inc., Advanced Book Program, Reading, Mass., 1978. |
[2] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978. |
[3] |
Z. Białynicka-Birula, Solitary waves in Born-Infeld electrodynamics, Bull. Acad. Pol. Sci., Sér. Sci. Phys. Astr., 27(1) (1979), 41-44. |
[4] |
H.-H. von Borzeszkowski and H.-J. Treder, Classical gravity and quantum matter fields in unified field theory, General Relativity and Gravitation, 28 (1996), 1-14. |
[5] |
H.-H. von Borzeszkowski and H.-J. Treder, Mach-Einstein doctrine and general relativity, Foundations of Physics, 26 (1996), 929-942.
doi: 10.1007/BF02148835. |
[6] |
H.-H. von Borzeszkowski and H.-J. Treder, Implications of Mach's principle: Dark matter and observable gravitions, in "Causality in Modern Physics" (eds. G. Hunter, et al) (Toronto, ON, 1997), Fund. Theories Phys., 97, Kluwer Academic Publishers, Dordrecht, (1998), 155-163. |
[7] |
H.-H. von Borzeszkowski and H.-J. Treder, Dark matter versus Mach's principle, Foundations of Physics, 28 (1998), 273-290.
doi: 10.1023/A:1018756904277. |
[8] |
G. Capriz, "Continua with Microstructure," Springer Tracts in Natural Philosophy, 35, Springer Verlag, New York, 1989. |
[9] |
G. Capriz and P. Giovine, On microstructural inertia,, Mathematical Models and Methods in Applied Sciences, 7 (1997), 211-216.
doi: 10.1142/S021820259700013X. |
[10] |
G. Capriz and P. M. Mariano, Symmetries and Hamiltonian formalism for complex materials, J. of Elasticity, 72 (2003), 57-70.
doi: 10.1023/B:ELAS.0000018775.44668.07. |
[11] |
F. J. Dyson, Dynamics of a spinning gas cloud, J. of Math. and Mech., 18 (1968), 91. |
[12] |
A. C. Eringen, "Nonlinear Theory of Continuous Media," McGraw-Hill Book Company, New York-Toronto-London, 1962. |
[13] |
A. C. Eringen, Nonlinear theory of micro elastic solids. Part I and II, Int. J. Eng. Sci., 1964. |
[14] |
A. C. Eringen, Mechanics of Micromorphic Continua, in "Proceedings of the IUTAM Symposium on Mechanics of Generalized Continua, Freudenstadt and Stuttgart, 1967" (ed. E. Kröner), 18, Springer, Berlin-Heidelberg-New York, (1968), 18-33. |
[15] |
A. C. Eringen, ed., "Continuum Physics. Vol. I. Mathematics," Academic Press, New York-London, 1971; Vol. II, 1975. |
[16] |
P. Godlewski, Generally-covariant and $GL(n,\mathbbR)$-invariant model of field of linear frames interacting with complex scalar field, Rep. Math. Phys., 38 (1996), 29-44.
doi: 10.1016/0034-4877(96)87676-2. |
[17] |
P. Godlewski, Generally covariant and $GL(n,\mathbbR)$-invariant model of field of linear frames interacting with a multiplet of complex scalar fields, Rep. Math. Phys., 40 (1997), 71-90.
doi: 10.1016/S0034-4877(97)85619-4. |
[18] |
, F. W. Hehl, G. D. Kerlick and P. Van der Heyde,, Physical Review D, 10 (1974).
doi: 10.1103/PhysRevD.10.1066. |
[19] |
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. |
[20] |
K. E. Hellwig and B. Wegner, "Mathematik und Theoretische Physik. Ein Integrierter Grundkurs für Physiker und Mathematiker," Vol. I, II, Walter de Gruyter, Berlin-New York, 1992/93. |
[21] |
S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," Vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. |
[22] |
L. D. Landau and E. M. Lifshitz, "The Classical Theory of Fields. Vol. 2," 4th edition, Butterworth-Heinemann, Oxford, 1975. |
[23] |
C. K. Möller, Energy-momentum complex in the general relativity theory, Danske Vidensk. Selsk, Mat-Fys Meddr., 31 (1959), 14. |
[24] |
C. Pellegrini and J. Plebański, Tetrad fields and gravitational fields, Mat.-Fys. Skr. Dan. Vid. Selsk, 2 (1963), 39 pp. |
[25] |
M. B. Rubin, On the theory of a Cosserat point and its application to the numerical solution of continuum problems, ASME J. Appl. Mech., 52 (1985), 368-372.
doi: 10.1115/1.3169055. |
[26] |
M. B. Rubin, On the numerical solution of one dimensional continuum problems using the theory of Cosserat point, ASME J. Appl. Mech., 52 (1985), 373-378.
doi: 10.1115/1.3169056. |
[27] |
M. B. Rubin, Free vibration of a rectangular parallelepiped using the theory of a Cosserat point, ASME J. Appl. Mech., 53 (1986), 45-50.
doi: 10.1115/1.3171736. |
[28] |
J. J. Sławianowski, "Geometry of Phase Spaces," A Wiley-Interscience Publication, John Wiley & Sons, Chichester, PWN--Polish Scientific Publishers, Warsaw, 1991. |
[29] |
J. J. Sławianowski, $GL(n,\mathbbR)$ as a candidate for fundamental symmetry in field theory, Nuovo Cimento B (11), 106 (1991), 645-668. |
[30] |
J. J. Sławianowski, Internal geometry, general covariance and generalized Born-Infeld models. Part I. Scalar fields, Arch. Mech., 46 (1994), 375-397. |
[31] |
J. J. Sławianowski, Search for fundamental models with affine symmetry: some results, some hypotheses and some essay, in "Geometry, Integrability and Quantization" (eds. I. Mladenov and A. Hirschfeld), Softex, Sofia, (2005), 126-172. |
[32] |
J. J. Sławianowski, Teleparallelism, modified born-infeld nonlinearity and space-time as a micromorphic ether, in "Topics in Mathematical Physics, General Relativity and Cosmology in Honor of Jerzy Plebański" (eds. Hugo Garcia-Compean, Bogdan Mielnik, Merced Montesinos and Maciej Przanowski), World Scientific Publ., Hackensack, New Jersey, (2006), 441-451. |
[33] |
J. J. Sławianowski, Quantization of affine bodies. Theory and applications in mechanics of structured media, in "Material Substructures in Complex Bodies: From Atomic Level to Continuum" (eds. G. Capriz and P. M. Mariano), Elsevier, Amsterdam-Heidelber-London-New York-Oxford-Paris-San Diego-San Francisco-Singapore-Sydney-Tokyo, (2007), 80-162. |
[34] |
J. J. Sławianowski, Geometrically implied nonlinearities in mechanics and field theory, in "Geometry, Integrability and Quantization" (eds. I. Mladenov and M. de Leon), Softex, Sofia, (2007), 48-118. |
[35] |
J. J. Sławianowski and V. Kovalchuk, Search for the geometrodynamical gauge group. Hypotheses and some results, in "Geometry, Integrability and Quantization" (ed. I. Mladenov), Softex, Sofia, (2008), 66-132. |
[36] |
J. J. Sławianowski, V. Kovalchuk, A. Sławianowska, B. Gołubowska, A. Martens, E. E. Rożko and Z. J. Zawistowski, Affine symmetry in mechanics of collective and internal modes. Part I. Classical models, Rep. on Math. Phys., 54 (2004), 373-427.
doi: 10.1016/S0034-4877(04)80026-0. |
[37] |
J. J. Sławianowski, V. Kovalchuk, A. Sławianowska, B. Gołubowska, A. Martens, E. E. Rożko and Z. J. Zawistowski, Affine symmetry in mechanics of collective and internal modes. Part II. Quantum models, Rep. on Math. Phys., 55 (2005), 1-46.
doi: 10.1016/S0034-4877(05)80002-3. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, revised and enlarged, With the assistance of Tudor Raţiu and Richard Cushman, Benjamin-Cummings Publishing Company, Inc., Advanced Book Program, Reading, Mass., 1978. |
[2] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978. |
[3] |
Z. Białynicka-Birula, Solitary waves in Born-Infeld electrodynamics, Bull. Acad. Pol. Sci., Sér. Sci. Phys. Astr., 27(1) (1979), 41-44. |
[4] |
H.-H. von Borzeszkowski and H.-J. Treder, Classical gravity and quantum matter fields in unified field theory, General Relativity and Gravitation, 28 (1996), 1-14. |
[5] |
H.-H. von Borzeszkowski and H.-J. Treder, Mach-Einstein doctrine and general relativity, Foundations of Physics, 26 (1996), 929-942.
doi: 10.1007/BF02148835. |
[6] |
H.-H. von Borzeszkowski and H.-J. Treder, Implications of Mach's principle: Dark matter and observable gravitions, in "Causality in Modern Physics" (eds. G. Hunter, et al) (Toronto, ON, 1997), Fund. Theories Phys., 97, Kluwer Academic Publishers, Dordrecht, (1998), 155-163. |
[7] |
H.-H. von Borzeszkowski and H.-J. Treder, Dark matter versus Mach's principle, Foundations of Physics, 28 (1998), 273-290.
doi: 10.1023/A:1018756904277. |
[8] |
G. Capriz, "Continua with Microstructure," Springer Tracts in Natural Philosophy, 35, Springer Verlag, New York, 1989. |
[9] |
G. Capriz and P. Giovine, On microstructural inertia,, Mathematical Models and Methods in Applied Sciences, 7 (1997), 211-216.
doi: 10.1142/S021820259700013X. |
[10] |
G. Capriz and P. M. Mariano, Symmetries and Hamiltonian formalism for complex materials, J. of Elasticity, 72 (2003), 57-70.
doi: 10.1023/B:ELAS.0000018775.44668.07. |
[11] |
F. J. Dyson, Dynamics of a spinning gas cloud, J. of Math. and Mech., 18 (1968), 91. |
[12] |
A. C. Eringen, "Nonlinear Theory of Continuous Media," McGraw-Hill Book Company, New York-Toronto-London, 1962. |
[13] |
A. C. Eringen, Nonlinear theory of micro elastic solids. Part I and II, Int. J. Eng. Sci., 1964. |
[14] |
A. C. Eringen, Mechanics of Micromorphic Continua, in "Proceedings of the IUTAM Symposium on Mechanics of Generalized Continua, Freudenstadt and Stuttgart, 1967" (ed. E. Kröner), 18, Springer, Berlin-Heidelberg-New York, (1968), 18-33. |
[15] |
A. C. Eringen, ed., "Continuum Physics. Vol. I. Mathematics," Academic Press, New York-London, 1971; Vol. II, 1975. |
[16] |
P. Godlewski, Generally-covariant and $GL(n,\mathbbR)$-invariant model of field of linear frames interacting with complex scalar field, Rep. Math. Phys., 38 (1996), 29-44.
doi: 10.1016/0034-4877(96)87676-2. |
[17] |
P. Godlewski, Generally covariant and $GL(n,\mathbbR)$-invariant model of field of linear frames interacting with a multiplet of complex scalar fields, Rep. Math. Phys., 40 (1997), 71-90.
doi: 10.1016/S0034-4877(97)85619-4. |
[18] |
, F. W. Hehl, G. D. Kerlick and P. Van der Heyde,, Physical Review D, 10 (1974).
doi: 10.1103/PhysRevD.10.1066. |
[19] |
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. |
[20] |
K. E. Hellwig and B. Wegner, "Mathematik und Theoretische Physik. Ein Integrierter Grundkurs für Physiker und Mathematiker," Vol. I, II, Walter de Gruyter, Berlin-New York, 1992/93. |
[21] |
S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," Vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. |
[22] |
L. D. Landau and E. M. Lifshitz, "The Classical Theory of Fields. Vol. 2," 4th edition, Butterworth-Heinemann, Oxford, 1975. |
[23] |
C. K. Möller, Energy-momentum complex in the general relativity theory, Danske Vidensk. Selsk, Mat-Fys Meddr., 31 (1959), 14. |
[24] |
C. Pellegrini and J. Plebański, Tetrad fields and gravitational fields, Mat.-Fys. Skr. Dan. Vid. Selsk, 2 (1963), 39 pp. |
[25] |
M. B. Rubin, On the theory of a Cosserat point and its application to the numerical solution of continuum problems, ASME J. Appl. Mech., 52 (1985), 368-372.
doi: 10.1115/1.3169055. |
[26] |
M. B. Rubin, On the numerical solution of one dimensional continuum problems using the theory of Cosserat point, ASME J. Appl. Mech., 52 (1985), 373-378.
doi: 10.1115/1.3169056. |
[27] |
M. B. Rubin, Free vibration of a rectangular parallelepiped using the theory of a Cosserat point, ASME J. Appl. Mech., 53 (1986), 45-50.
doi: 10.1115/1.3171736. |
[28] |
J. J. Sławianowski, "Geometry of Phase Spaces," A Wiley-Interscience Publication, John Wiley & Sons, Chichester, PWN--Polish Scientific Publishers, Warsaw, 1991. |
[29] |
J. J. Sławianowski, $GL(n,\mathbbR)$ as a candidate for fundamental symmetry in field theory, Nuovo Cimento B (11), 106 (1991), 645-668. |
[30] |
J. J. Sławianowski, Internal geometry, general covariance and generalized Born-Infeld models. Part I. Scalar fields, Arch. Mech., 46 (1994), 375-397. |
[31] |
J. J. Sławianowski, Search for fundamental models with affine symmetry: some results, some hypotheses and some essay, in "Geometry, Integrability and Quantization" (eds. I. Mladenov and A. Hirschfeld), Softex, Sofia, (2005), 126-172. |
[32] |
J. J. Sławianowski, Teleparallelism, modified born-infeld nonlinearity and space-time as a micromorphic ether, in "Topics in Mathematical Physics, General Relativity and Cosmology in Honor of Jerzy Plebański" (eds. Hugo Garcia-Compean, Bogdan Mielnik, Merced Montesinos and Maciej Przanowski), World Scientific Publ., Hackensack, New Jersey, (2006), 441-451. |
[33] |
J. J. Sławianowski, Quantization of affine bodies. Theory and applications in mechanics of structured media, in "Material Substructures in Complex Bodies: From Atomic Level to Continuum" (eds. G. Capriz and P. M. Mariano), Elsevier, Amsterdam-Heidelber-London-New York-Oxford-Paris-San Diego-San Francisco-Singapore-Sydney-Tokyo, (2007), 80-162. |
[34] |
J. J. Sławianowski, Geometrically implied nonlinearities in mechanics and field theory, in "Geometry, Integrability and Quantization" (eds. I. Mladenov and M. de Leon), Softex, Sofia, (2007), 48-118. |
[35] |
J. J. Sławianowski and V. Kovalchuk, Search for the geometrodynamical gauge group. Hypotheses and some results, in "Geometry, Integrability and Quantization" (ed. I. Mladenov), Softex, Sofia, (2008), 66-132. |
[36] |
J. J. Sławianowski, V. Kovalchuk, A. Sławianowska, B. Gołubowska, A. Martens, E. E. Rożko and Z. J. Zawistowski, Affine symmetry in mechanics of collective and internal modes. Part I. Classical models, Rep. on Math. Phys., 54 (2004), 373-427.
doi: 10.1016/S0034-4877(04)80026-0. |
[37] |
J. J. Sławianowski, V. Kovalchuk, A. Sławianowska, B. Gołubowska, A. Martens, E. E. Rożko and Z. J. Zawistowski, Affine symmetry in mechanics of collective and internal modes. Part II. Quantum models, Rep. on Math. Phys., 55 (2005), 1-46.
doi: 10.1016/S0034-4877(05)80002-3. |
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