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.
Research Supported in Part by the National Science Foundation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Klein, F.: Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, vol. 2, ch. 2. Berlin: Springer 1927.
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 1957
A detailed discussion is given by A. Wintner, The analytical foundations of celestial mechanics, ch. V. Princeton: Princeton University Press 1947.
Noether, E.: Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 235 (1918).
A simplified discussion with examples is given by E. L. Hill, Rev. Modern Phys. 23, 253 (1951)
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.
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.
For a discussion of the historical development of these concepts see Mary B. Hesse, Forces and fields. London: Thomas Nelson & Sons Ltd 1961.
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.
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,
P. Hayas, in: Statistical mechanics of equilibrium and non-equilibrium, ed. by J. Meix-ner, p. 1. Amsterdam: North-Holland Publ. Co. 1965.
Havas, P.: Rev. Modern Phys. 36, 938 (1964), and references given there.
First found for point charges by A.D. Fokker, Z. Physik 58, 386 (1929); compare also
K. Schwarzschild, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 126 (1903).
For the case of mesic interactions, see P. Hayas, Phys. Rev. 87,309 (1952), and references given there.
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).
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.
Schütz, J.R.: Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 110 (1897).
Königsberger, L.: Die Principien der Mechanik. Leipzig: B.G.Teubner 1901.
Eisenbud, L., Wigner, E.P.: Proc. Nat. Acad. Sci. U.S.A. 27, 281 (1941).
Houtappel, R. M. F.,Van Dam, H., Wigner, E. P.: Rev. Modern Phys. 37,595 (1965).
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.
Havas, P. (to be published).
The presentation of this section substantially is that given by P. Havas and J. Stachel, Phys. Rev. 185, 1636 (1969).
For an example with infinitely many conservation laws see H. Steudel, Z. Naturforsch. 21A, 1826 (1966).
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.
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.
This result is implicitly contained in H. A. Kastrup, Nucl Phys. 58,561 (1964).
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.
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].
Havas, P., Plebański, J.: Bull. Amer. Phys. Soc. 5, 433 (1960); see also P. Havas, Ref. [8].
VanDam, H., Wigner, E.P.: Phys. Rev. 138, B 1576 (1965); 142, 838 (1966).
For the case of Coulomb’s law this will be discussed elsewhere.
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.
Dettman, J.W., Schild, A.: Phys. Rev. 95, 1057 (1954).
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).
Havas, P.: Phys. Rev. 87, 898 (1952).
Fokker, A.D.: Physica 9, 33 (1929).
In electrodynamics this was shown first by J. Frenkel, Z. Physik 32, 518 (1925)
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].
Wheeler, J. A., Feynman, R.P.: Rev. Modern Phys. 17, 157 (1945);
see also P. Havas, Phys. Rev. 74, 456 (1948); 86, 974 (1952).
Havas, P.: Phys. Rev. 91, 997 (1953), and a paper in preparation.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1971 Springer-Verlag Berlin · Heidelberg
About this chapter
Cite this chapter
Havas, P. (1971). Galilei-and Lorentz-Invariant Particle Systems and their Conservation Laws. In: Bunge, M. (eds) Problems in the Foundations of Physics. Studies in the Foundations, Methodology and Philosophy of Science, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80624-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-80624-7_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-80626-1
Online ISBN: 978-3-642-80624-7
eBook Packages: Springer Book Archive