Advertisement

Annali di Matematica Pura ed Applicata

, Volume 57, Issue 1, pp 339–403 | Cite as

The Riemannian geometry of physical systems of curves

  • John De Cicco
Article
  • 27 Downloads

Summary

The properties of a physical system Sk where k ≠−1, of ∞2n−1 trajectories C. in a Riemannian space Vn are developed. The intrinsic differential equations and the equations of Lagrange, of a physical system Sk, are derived. The Lagrangian function L and the Hamiltonian function H, are studied in the conservative case. Also included are systems of the type (G), curvature trajectories, and natural families. The Appell transformation T of a dynamical system S 0 in a Riemannian space Vn, is obtained. Finally, contact transformations and the transformation theory of a physical system Sk where k ≠−1, are considered in detail.

Keywords

Differential Equation Dynamical System Physical System Lagrangian Function Riemannian Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    Kasner andDe Cicco,Geometrical properties of physical curves in space of n dimensions, Revists de Matematica y Fisica Teorica de la Universidad Nacional del Tucuman, (Argentina), 8, 127–137, 1951.Google Scholar
  2. [2]
    These were studied originally byKasner in the Euclidean plane. See his Princeton Colloquium Lectures, pages 8 and 9.Google Scholar
  3. [3]
    Eisenhart,Riemannian geometry, Princeton University Press 1949.Google Scholar
  4. [4]
    Whittaker,A treatise on the analytical dynamics of particles and rigid bodies, Fourth edition, Dover Publications, 1944.Google Scholar
  5. [5]
    De Cicco,Conservative physical families of curves on a surface, Revists de « Matematica y Fisica Teorica de la Universidad Nacional del Tucuman », (Argentina), 9, 23–36, 1952.zbMATHGoogle Scholar
  6. [6]
    Kasner,Differential equations of the type; y′''=Gy′' + Hy′' 2, Proceedings of the National Academy of Sciences, 28, 333–338, 1942.zbMATHMathSciNetGoogle Scholar
  7. [7]
    Kasner andMittleman,A general theorem of the initial curvature of dynamical trajectories, Proceedings of the National Academy of Sciences, 28, 48–52, 1942.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Kasner andDe Cicco,A generalized theory of dynamical trajectories, Transactions of the « American Mathematical Society », 54, 23–38, 1943.CrossRefMathSciNetzbMATHGoogle Scholar
  9. [9]
    Kasner,Dynamical trajectories and curvature trajectories, « Bulletin of the American Mathematical Society », 40, 449–455, 1934.zbMATHMathSciNetGoogle Scholar
  10. [10]
    For an extension of this theorem ofKasner to space, seeDe Cicco,Dynamical and curvature trajectories in space, Transactions of the American Society, 57, 270–286, 1945.CrossRefzbMATHGoogle Scholar
  11. [11]
    Kasner andDe Cicco,The geometry of velocity systems, « Bulletin of the American Mathematical Society, 49, 236–245 », 1943.zbMATHMathSciNetGoogle Scholar
  12. [12]
    Kasner,Natural families of trajectories: Conservative fields of force, Transactions of the « American Mathematical Society », 10, 201–219, 1909. Also seeKasner,The theorem of Thompson and Tait, and natural families of trajectories, Transactions of the « American Mathematical Society », 11, 121–140, 1910.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    Kasner andDe Cicco,Physical families in conservative fields of force, Proceedings of the « National Academy of Sciences, 35, 419–422, 1949.zbMATHGoogle Scholar
  14. [14]
    Kasner,The infinitesimal contact transformations of mechanics, « Bulletin of the American Mathematical Society, 16, 408–412, 1910.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Eisenhart,Continuous groups of transformations, « Princeion University Press, 1933 ».Google Scholar
  16. [16]
    De Cicco,New proofs of the theorems of Kasner concerning the infinitesimal contact transformations of mechanics, « Journal of Mathematics and Physics, 26, 104–109 », 1947.zbMATHMathSciNetGoogle Scholar
  17. [17]
    Eisenhart,Non-Riemannian geometry, « American Mathematical Society » Colloquium Publications, 1927.Google Scholar
  18. [18]
    Kasner andDe Cicco,Generalization of Appell's transformation, « Journal of Mathematics and Physics », 27, 262–269, 1949.zbMATHGoogle Scholar
  19. [19]
    —— ——,Transformation theory of physical curves, Proceedings of the « National Academy of Sciences, 33, 338–342, 1947.zbMATHGoogle Scholar

Copyright information

© Nicola Zanichelli Editore 1962

Authors and Affiliations

  • John De Cicco
    • 1
  1. 1.ChicagoU.S.A.

Personalised recommendations