Far Eastern Mathematical Journal

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On covariant form of the momentum balance equation for perfect fluid


Gudimenko A. I., Guzev M. A.

2015, issue 1, Đ. 41-52


Abstract
The apparatus of differential geometry is used to represent the momentum balance equation for perfect fluid in Ó form that is invariant under the time-dependent coordinate transformations. The motion of fluid is described in the framework of four-dimensional formalism when the space-time is represented as a bundle over the time axis $\mathbb R$. Applications of the obtained formulation are discussed.

Keywords:
continuum mechanics, momentum conservation law, fiberbundles, covariant formulation

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References

[1] C. Truesdell, W. Noll, The non-linear eld theories of mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 2004, 603 pp.
[2] W. Noll, Five contributions to natural philosophy, 2004, http://www.math.cmu.edu/wn0g/noll.
[3] C. Truesdell, R. Toupin, ""The classical eld theories"", In: Encyclopedia of Physics, ed. S. Flugge, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1960.
[4] J. E. Marsden, T. J. R. Hughes, Mathematical foundations of elasticity, Dover, New York, 1983.
[5] G. Romano, R. Barretta, Continuum mechanics on manifolds, University of Naples Federico II, Naples, Italy, 2014, http://wpage.unina.it/romano.
[6] G. Romano, R. Barretta and M. Diaco, ""Geometric continuum mechanics"", Meccanica, 49:1, (2014), 111-133.
[7] A. I. Gudimenko, M. A. Guzev, ôOb invariantnoj forme zapisi zakona sohranenija massyö, Dal'nevost. matem. zhurn., 14:1 (2014), 33ľ40.
[8] A. I. Gudimenko, M. A. Guzev, ôGeometricheskie aspekty izuchenija zakona sohranenijamassyö, Dal'nevost. matem. zhurn., 14:2 (2014), 173ľ190.
[9] D. Saunders, The Geometry of Jet Bundles, Cambridge Univ. Press, Cambridge, 1989.
[10] G. A. Sardanashvili, Sovremennye metody teorii polja. 2. Geometrija i klassicheskaja mehanika, URSS, Moskva, 1998.
[11] L. Mangiarotti and G. Sardanashvily, Connections in classical and quantum field theory,World Scientific, Singapore, NewJersey, London, Hong Kong, 2000.
[12] G. Lamb, Gidrodinamika, Gostehizdat, Moskva, 1947.
[13] V. I. Arnol'd, V. V. Kozlov, A. I. Nejshtadt, Matematicheskie aspekty klassicheskoj i nebesnoj mehaniki, Sovremennye problemy matematiki. Fundamental'nye naprvlenija. T. 3, VINITI, Moskva, 1985.
[14] B. Schutz, Geometrical methods of mathematical physics, Cambridge Univ. Press, Cambridge, 1980.
[15] F. I. Dolzhanskij, Lekcii po geofizicheskoj gidrodinamiki, IBM RAN, Moskva, 2006.
[16] L. D. Landau, E. M. Lifshic, Teoreticheskaja fizika. T. VI. Gidrodinamika, Fizmatlit, Moskva, 2001.
[17] B. A. Dubrovin, S. P. Novikov, A.T. Fomenko, Sovremennaja geometrija. Metody i prilozhenija, Nauka. Gl. red. fiz.-mat. lit., Moskva, 1986.
[18] L. G. Lojcjanskij, Mehanika zhidkosti i gaza: Ucheb. dlja vuzov., Drofa, Moskva, 2003.
[19] D. V. Sivuhin, Obshhij kurs fiziki. T. I. Mehanika, Fizmatlit. Izd-vo MFTI, Moskva, 2005.
[20] S. K. Godunov, E. I. Romenskij, Jelementy mehaniki sploshnyh sred i zakony sohranenija, Nauchnaja kniga, Novosibirsk, 1998.
[21] F. Uorner, Osnovy teorii gladkih mnogoobrazij i grupp Li, Mir, Moskva, 1987.
[22] I. Kol ar and P. Michor and J. Slovak, Natural operations in differential geometry,Springer-Verlag, Berlin, Heidelberg, New York, 1993.

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