Continuum Mechanics and Thermodynamics

, Volume 31, Issue 3, pp 627–638 | Cite as

The integral theorem of generalized virial in the relativistic uniform model

  • Sergey G. Fedosin
Original Article


In the relativistic uniform model for continuous medium, the integral theorem of generalized virial is derived, in which generalized momenta are used as particles’ momenta. This allows us to find exact formulas for the radial component of the velocity of typical particles of the system and for their root-mean-square speed, without using the notion of temperature. The relation between the theorem and the cosmological constant, characterizing the physical system under consideration, is shown. The difference is explained between the kinetic energy and the energy of motion, the value of which is equal to half the sum of the Lagrangian and the Hamiltonian. This difference is due to the fact that the proper fields of each particle have mass–energy, which makes an additional contribution into the kinetic energy. As a result, the total energy of motion of particles and fields is obtained.


Generalized virial theorem Relativistic uniform model Cosmological constant Energy of motion Kinetic energy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, Boston (1980)zbMATHGoogle Scholar
  2. 2.
    Ganghoffer, J., Rahouadj, R.: On the generalized virial theorem for systems with variable mass. Continuum Mech. Thermodyn. 28, 443–463 (2016). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fock, V.: Bemerkung zum Virialsatz. Zeitschrift für Physik A. 63(11), 855–858 (1930). ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Parker, E.N.: Tensor virial equations. Phys. Rev. 96(6), 1686–1689 (1954). 96.1686Google Scholar
  5. 5.
    Fedosin, S.G.: The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept. Contin. Mech. Thermodyn. 29(2), 361–371 (2016). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Landau, L.D., Lifschitz, E.M.: Mechanics. Course of Theoretical Physics. Vol. 1 (3rd ed.). London: Pergamon. ISBN 0-08-021022-8 (1976)Google Scholar
  7. 7.
    Fedosin, S.G.: About the cosmological constant, acceleration field, pressure field and energy. Jordan J. Phys. 9(1), 1–30 (2016). ADSGoogle Scholar
  8. 8.
    Fedosin, S.G.: The procedure of finding the stress-energy tensor and vector field equations of any form. Adv. Stud. Theor. Phys. 8, 771–779 (2014). CrossRefGoogle Scholar
  9. 9.
    Fedosin, S.G.: The integral energy–momentum 4-vector and analysis of 4/3 problem based on the pressure field and acceleration field. Am. J. Mod. Phys. 3(4), 152–167 (2014). CrossRefGoogle Scholar
  10. 10.
    Fedosin, S.G.: Relativistic energy and mass in the weak field limit. Jordan J. Phys. 8(1), 1–16 (2015). Google Scholar
  11. 11.
    Fedosin, S.G.: 4/3 problem for the gravitational field. Adv. Phys. Theor. Appl. 23, 19–25 (2013). Google Scholar
  12. 12.
    Fedosin, S.G.: Estimation of the physical parameters of planets and stars in the gravitational equilibrium model. Canad. J. Phys. 94(4), 370–379 (2016). ADSCrossRefGoogle Scholar
  13. 13.
    Reif, F.: Fundamentals of Statistical and Thermal Physics. Long Grove, IL: Waveland Press, Inc. ISBN 1-57766-612-7 (2009)Google Scholar
  14. 14.
    Fedosin, S.G.: The Hamiltonian in covariant theory of gravitation. Adv. Nat. Sci. 5(4), 55–75 (2012). Google Scholar
  15. 15.
    Dirac P.A.M.: General Theory of Relativity. Princeton University Press, quick presentation of the bare essentials of GTR. ISBN 0-691-01146-X (1975)Google Scholar
  16. 16.
    Denisov, V.I., Logunov, A.A.: The inertial mass defined in the general theory of relativity has no physical meaning. Theoretical and Mathematical Physics 51(2), 421–426 (1982). ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    Khrapko, R.I.: The truth about the energy–momentum tensor and pseudotensor. ISSN 0202-2893, Gravitation and Cosmology, 20(4), 264–273 (2014). Pleiades Publishing, Ltd., 2014.
  18. 18.
    Snider, R.F.: Conversion between kinetic energy and potential energy in the classical nonlocal Boltzmann equation. J. Stat. Phys. 80, 1085–1117 (1995). ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Sergey G. Fedosin
    • 1
  1. 1.PermRussia

Personalised recommendations