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March  2012, 17(2): 699-733. doi: 10.3934/dcdsb.2012.17.699

Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics

Jan J. Sławianowski 1, , Vasyl Kovalchuk 1, , Agnieszka Martens 1, , Barbara Gołubowska 1, and  Ewa E. Rożko 1,

1. 

Institute of Fundamental Technological Research, Polish Academy of Sciences, 5B, Pawińskiego str., 02-106 Warsaw, Poland, Poland, Poland, Poland, Poland

Received  November 2010 Revised  April 2011 Published  December 2011

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: tetrad models and micromorphic continua., dynamical symmetries, general relativity, affine bodies, Essential nonlinearity.
Mathematics Subject Classification: Primary: 37J15, 53D05, 74A60 83D05, 83E05; Secondary: 53A15, 53D2.
Citation: Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699
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.  Google Scholar

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978.  Google Scholar

[3]

Z. Białynicka-Birula, Solitary waves in Born-Infeld electrodynamics, Bull. Acad. Pol. Sci., Sér. Sci. Phys. Astr., 27(1) (1979), 41-44.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. Capriz, "Continua with Microstructure," Springer Tracts in Natural Philosophy, 35, Springer Verlag, New York, 1989.  Google Scholar

[9]

G. Capriz and P. Giovine, On microstructural inertia,, Mathematical Models and Methods in Applied Sciences, 7 (1997), 211-216. doi: 10.1142/S021820259700013X.  Google Scholar

[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.  Google Scholar

[11]

F. J. Dyson, Dynamics of a spinning gas cloud, J. of Math. and Mech., 18 (1968), 91. Google Scholar

[12]

A. C. Eringen, "Nonlinear Theory of Continuous Media," McGraw-Hill Book Company, New York-Toronto-London, 1962.  Google Scholar

[13]

A. C. Eringen, Nonlinear theory of micro elastic solids. Part I and II, Int. J. Eng. Sci., 1964. Google Scholar

[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. Google Scholar

[15]

A. C. Eringen, ed., "Continuum Physics. Vol. I. Mathematics," Academic Press, New York-London, 1971; Vol. II, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

, F. W. Hehl, G. D. Kerlick and P. Van der Heyde,, Physical Review D, 10 (1974).  doi: 10.1103/PhysRevD.10.1066.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," Vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963.  Google Scholar

[22]

L. D. Landau and E. M. Lifshitz, "The Classical Theory of Fields. Vol. 2," 4th edition, Butterworth-Heinemann, Oxford, 1975. Google Scholar

[23]

C. K. Möller, Energy-momentum complex in the general relativity theory, Danske Vidensk. Selsk, Mat-Fys Meddr., 31 (1959), 14. Google Scholar

[24]

C. Pellegrini and J. Plebański, Tetrad fields and gravitational fields, Mat.-Fys. Skr. Dan. Vid. Selsk, 2 (1963), 39 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. J. Sławianowski, "Geometry of Phase Spaces," A Wiley-Interscience Publication, John Wiley & Sons, Chichester, PWN--Polish Scientific Publishers, Warsaw, 1991.  Google Scholar

[29]

J. J. Sławianowski, $GL(n,\mathbbR)$ as a candidate for fundamental symmetry in field theory, Nuovo Cimento B (11), 106 (1991), 645-668.  Google Scholar

[30]

J. J. Sławianowski, Internal geometry, general covariance and generalized Born-Infeld models. Part I. Scalar fields, Arch. Mech., 46 (1994), 375-397.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978.  Google Scholar

[3]

Z. Białynicka-Birula, Solitary waves in Born-Infeld electrodynamics, Bull. Acad. Pol. Sci., Sér. Sci. Phys. Astr., 27(1) (1979), 41-44.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. Capriz, "Continua with Microstructure," Springer Tracts in Natural Philosophy, 35, Springer Verlag, New York, 1989.  Google Scholar

[9]

G. Capriz and P. Giovine, On microstructural inertia,, Mathematical Models and Methods in Applied Sciences, 7 (1997), 211-216. doi: 10.1142/S021820259700013X.  Google Scholar

[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.  Google Scholar

[11]

F. J. Dyson, Dynamics of a spinning gas cloud, J. of Math. and Mech., 18 (1968), 91. Google Scholar

[12]

A. C. Eringen, "Nonlinear Theory of Continuous Media," McGraw-Hill Book Company, New York-Toronto-London, 1962.  Google Scholar

[13]

A. C. Eringen, Nonlinear theory of micro elastic solids. Part I and II, Int. J. Eng. Sci., 1964. Google Scholar

[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. Google Scholar

[15]

A. C. Eringen, ed., "Continuum Physics. Vol. I. Mathematics," Academic Press, New York-London, 1971; Vol. II, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

, F. W. Hehl, G. D. Kerlick and P. Van der Heyde,, Physical Review D, 10 (1974).  doi: 10.1103/PhysRevD.10.1066.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," Vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963.  Google Scholar

[22]

L. D. Landau and E. M. Lifshitz, "The Classical Theory of Fields. Vol. 2," 4th edition, Butterworth-Heinemann, Oxford, 1975. Google Scholar

[23]

C. K. Möller, Energy-momentum complex in the general relativity theory, Danske Vidensk. Selsk, Mat-Fys Meddr., 31 (1959), 14. Google Scholar

[24]

C. Pellegrini and J. Plebański, Tetrad fields and gravitational fields, Mat.-Fys. Skr. Dan. Vid. Selsk, 2 (1963), 39 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. J. Sławianowski, "Geometry of Phase Spaces," A Wiley-Interscience Publication, John Wiley & Sons, Chichester, PWN--Polish Scientific Publishers, Warsaw, 1991.  Google Scholar

[29]

J. J. Sławianowski, $GL(n,\mathbbR)$ as a candidate for fundamental symmetry in field theory, Nuovo Cimento B (11), 106 (1991), 645-668.  Google Scholar

[30]

J. J. Sławianowski, Internal geometry, general covariance and generalized Born-Infeld models. Part I. Scalar fields, Arch. Mech., 46 (1994), 375-397.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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