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Galilei-and Lorentz-Invariant Particle Systems and their Conservation Laws

Chapter
Part of the Studies in the Foundations, Methodology and Philosophy of Science book series (FOUNDATION, volume 4)

Abstract

Newton’s mechanics of point particles has served as a model of physical theory for two centuries. In its usual formulation, the forces considered are two-body forces depending on the mutual separation of the particles at the same time. The third law implies that these forces are derivable from a potential also depending only on this separation, which allows a Lagrangean formulation of the theory. As was gradually recognized during the past century, this particular form of Newtonian mechanics is invariant under a ten-parameter group, the inhomogeneous Galilei group [1], and possesses ten integrals which are algebraic functions of the positions and velocities; one of these corresponds to the conservation of energy, three to the conservation of linear momentum, three to the conservation of angular momentum, and three express the uniform motion of the center of mass.

Keywords

Variational Principle Lorentz Group Newtonian Mechanic World Line Modern Phys 
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References

  1. 1.
    Klein, F.: Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, vol. 2, ch. 2. Berlin: Springer 1927.Google Scholar
  2. 2.
    For an elementary derivation for the case of gravitational interaction see e. g. R. Kurth, Introduction to the mechanics of stellar systems, ch. III. New York: Pergamon Press 1957zbMATHGoogle Scholar
  3. A detailed discussion is given by A. Wintner, The analytical foundations of celestial mechanics, ch. V. Princeton: Princeton University Press 1947.zbMATHGoogle Scholar
  4. 3.
    Noether, E.: Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 235 (1918).ADSGoogle Scholar
  5. 4.
    A simplified discussion with examples is given by E. L. Hill, Rev. Modern Phys. 23, 253 (1951)ADSzbMATHCrossRefGoogle Scholar
  6. for a more general discussion see A. Trautman, in: Gravitation, ed. by L. Witten, ch. 5, New York-London: John Wiley & Sons 1962, or in: Brandeis summer institute in theoretical physics, 1964, vol. 1, ch. 7. Englewood Cliffs, N. J.: Prentice-Hall, Inc. 1965.Google Scholar
  7. 5.
    Bessel-Hagen, E.: Math. Ann. 84, 258 (1921). This paper includes a slightly more general formulation of Noether’s theorems than stated in Ref. [3]; it too is credited to Noether. We shall use this formulation throughout.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 6.
    For a discussion of the historical development of these concepts see Mary B. Hesse, Forces and fields. London: Thomas Nelson & Sons Ltd 1961.Google Scholar
  9. 7.
    For a brief discussion of the physical and conceptual problems see P. Havas, in: Proceedings of the 1964 international congress of logic, methodology and philosophy of science, p. 347. Amsterdam: North-Holland Publ. Co. 1965.Google Scholar
  10. 8.
    Surveys of recent work are given by D.G. Currie and T.F. Jordan in: Lectures in theoretical physics, vol. XA, p.91, New York: Gordon and Breach 1968,Google Scholar
  11. P. Hayas, in: Statistical mechanics of equilibrium and non-equilibrium, ed. by J. Meix-ner, p. 1. Amsterdam: North-Holland Publ. Co. 1965.Google Scholar
  12. 9.
    Havas, P.: Rev. Modern Phys. 36, 938 (1964), and references given there.MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 10.
    First found for point charges by A.D. Fokker, Z. Physik 58, 386 (1929); compare alsoADSCrossRefGoogle Scholar
  14. K. Schwarzschild, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 126 (1903).Google Scholar
  15. 11.
    For the case of mesic interactions, see P. Hayas, Phys. Rev. 87,309 (1952), and references given there.ADSCrossRefGoogle Scholar
  16. 12.
    Such a variational principle can be obtained also in a Lorentz-invariant approximation method in the general theory of relativity, as shown by P. Hayas and J. N. Goldberg, Phys. Rev. 128, 398 (1962).MathSciNetADSCrossRefGoogle Scholar
  17. 13.
    See P.G. Bergmann in the article on Special Relativity, part Bl, in: Encyclopedia of physics, ed. by S. Flügge, vol. IV. Berlin-Göttingen-Heidelberg: Springer 1962.Google Scholar
  18. 14.
    Schütz, J.R.: Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 110 (1897).Google Scholar
  19. 15.
    Königsberger, L.: Die Principien der Mechanik. Leipzig: B.G.Teubner 1901.Google Scholar
  20. 16.
    Eisenbud, L., Wigner, E.P.: Proc. Nat. Acad. Sci. U.S.A. 27, 281 (1941).ADSCrossRefGoogle Scholar
  21. 17.
    Houtappel, R. M. F.,Van Dam, H., Wigner, E. P.: Rev. Modern Phys. 37,595 (1965).MathSciNetADSCrossRefGoogle Scholar
  22. Houtappel, R. M F Arens, R.: j. Math. Phys. 1, 1341 (1966). n-body forces seem to have been suggested first by G.T. Fechner, Physikalische and Philosophische Atomlehre. Leipzig: H. Mendelssohn 1864.Google Scholar
  23. 18.
    Havas, P. (to be published).Google Scholar
  24. 19.
    The presentation of this section substantially is that given by P. Havas and J. Stachel, Phys. Rev. 185, 1636 (1969).MathSciNetADSCrossRefGoogle Scholar
  25. 20.
    For an example with infinitely many conservation laws see H. Steudel, Z. Naturforsch. 21A, 1826 (1966).Google Scholar
  26. 21.
    For the specific case of Newton’s law of gravitation, the method and results of this section are due to Bessel-Hagen [5] and can also be found in Hill [4]. The generalization to arbitrary potentials of the form (7) is trivial.Google Scholar
  27. 22.
    Anderson, C. M., Baeyer, H.C. von (in press). It should be noted that these authors obtained such conservation laws by considering scale transformations of the masses and coupling constants in addition to those of the coordinates, which is not necessary, as just demonstrated.Google Scholar
  28. 23.
    This result is implicitly contained in H. A. Kastrup, Nucl Phys. 58,561 (1964).MathSciNetCrossRefGoogle Scholar
  29. 24.
    In the following, corresponding formulae of the Lorentz- and the Galilei case will be designated by L and G, respectively; formulae without a letter hold for both cases.Google Scholar
  30. 25.
    It should be noted that the Galilean and Lorentzian Fjp (unlike the Fj p) have different dimensions; the same difference appears in the Uijof Eqs.(45) and the Vj of Eq.(48) below. The necessity of such a difference is discussed in Ref. [9].Google Scholar
  31. 26.
    Havas, P., Plebański, J.: Bull. Amer. Phys. Soc. 5, 433 (1960); see also P. Havas, Ref. [8].Google Scholar
  32. 27.
    VanDam, H., Wigner, E.P.: Phys. Rev. 138, B 1576 (1965); 142, 838 (1966).Google Scholar
  33. 28.
    For the case of Coulomb’s law this will be discussed elsewhere.Google Scholar
  34. 29.
    Approximately relativistic Lagrangeans (with correction terms up to order 1/c2) are discussed by P. Havas and J. Stachel (Ref. [19]), by H. Woodcock, Temple University Thesis, 1971, and in several papers in preparation by these authors.Google Scholar
  35. 30.
    Dettman, J.W., Schild, A.: Phys. Rev. 95, 1057 (1954).MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. 31.
    For Fokker’s principle in electrodynamics (Ref. [10]), the conservation law (56) was derived by J.A. Wheeler and R.P. Feynman, Rev. Mod. Phys. 21, 425 (1949).MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 32.
    Havas, P.: Phys. Rev. 87, 898 (1952).MathSciNetADSCrossRefGoogle Scholar
  38. 33.
    Fokker, A.D.: Physica 9, 33 (1929).Google Scholar
  39. 34.
    In electrodynamics this was shown first by J. Frenkel, Z. Physik 32, 518 (1925)ADSCrossRefGoogle Scholar
  40. J. L. Synge, Trans. Roy. Soc. Can. 34, 1 (1940); an alternate form of the energy-momentum tensor was suggested in Ref. [31]. The results were generalized to mesodynamics in Ref. [11].MathSciNetzbMATHGoogle Scholar
  41. 35.
    Wheeler, J. A., Feynman, R.P.: Rev. Modern Phys. 17, 157 (1945);ADSCrossRefGoogle Scholar
  42. see also P. Havas, Phys. Rev. 74, 456 (1948); 86, 974 (1952).ADSzbMATHCrossRefGoogle Scholar
  43. 36.
    Havas, P.: Phys. Rev. 91, 997 (1953), and a paper in preparation.MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  1. 1.Department of PhysicsTemple UniversityPhiladelphiaUSA

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