Skip to main content

Riemannian parallel translation in non-linear mechanics

  • Conference paper
  • First Online:
Global Analysis — Studies and Applications I

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1108))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnold V.I. Mathematical methods of classical mechanics. Moscow, 1974 (in Russian).

    Google Scholar 

  2. Bishop R.L. and Crittenden R.G. Geometry of manifolds.-Academic Press, N.Y.-London, 1964.

    MATH  Google Scholar 

  3. Bolotin V.R. Random vibrations of mechanical systems. Moscow, 1979 (in Russian).

    Google Scholar 

  4. Borisov Yu.F. Parallel translation along Hölder curves in a Riemannian space.-Dokl. Akad. Nauk SSSR, 1971, 197, No. 5 (in Russian).

    Google Scholar 

  5. Borisovich Yu.G., Gelman B.D., Myshkis A.D., and Obukhovsky V.V. Topological methods in the theory of fixed points of many-valued mappings.-Usp. Matem. Nauk, 1980, 35, No. 1 (in Russian).

    Google Scholar 

  6. Borisovič Ju. and Gliklih Ju.E. Fixed points of mappings of Banach manifolds and some applications.-Nonlinear analysis: Theory, Methods, and Applications, 1980, vol. 4, No. 1.

    Google Scholar 

  7. Warga J. Optimal control of differential and functional equations.-Acad. Press, N.Y.-London, 1972.

    MATH  Google Scholar 

  8. Vershik A.M. and Faddeev L.D. Lagrangian mechanics in invariant exposition.-in: Problems of theoretical physics. Leningrad, 1975, No. 2 (in Russian).

    Google Scholar 

  9. Gelig A.Kh., Leonov G.A., and Yakubovich V.A. Stability of nonlinear systems with a non-unique equilibrium state. Moscow, 1978 (in Russian).

    Google Scholar 

  10. Gelman B.D. and Gliklikh Yu.E. A two-point boundary-value problem in geometric mechanics with discontinuous forces.-Prikladnaya Matematika i Mekhanika, 1980, 43, No. 3 (in Russian).

    Google Scholar 

  11. Gikhman I.I. and Skorokhod A.V. The theory of random processes. Moscow, 1975, vol. 3 (in Russian).

    Google Scholar 

  12. Gikhman I.I. and Skorokhod A.V. Introduction to the theory of random processes. Moscow, 1977 (in Russian).

    Google Scholar 

  13. Gliklikh Yu.E. Integral operators on a manifold.-Trudy Matem. Fakulteta, Voronezh State Univ., 1971, No. 4 (in Russian).

    Google Scholar 

  14. Gliklikh Yu.E. On a certain generalization of the Hopf-theorem about geodesics.-Usp. Matem. Nauk, 1974, 29, No. 6 (in Russian).

    Google Scholar 

  15. Gliklikh Yu.E. A two-point boundary-value problem in geometric mechanics of systems with a bounded force field. Deposited at VINITI, June 6, 1977, No. 2217-77 (in Russian).

    Google Scholar 

  16. Gliklikh Yu.E. On a certain analogue of differential equations with discrete delay on a Riemannian manifold. Deposited at VINITI, March 20, 1980, No. 1089-80 (in Russian).

    Google Scholar 

  17. Gliklikh Yu.E. On systems of geometric mechanics with a delayed control force. Deposited at VINITI, October 14, 1980, No. 4384-80 (in Russian).

    Google Scholar 

  18. Gliklikh Yu.E. On equations with discrete delay on non-linear spaces.-in: Operator methods in non-linear analysis. Voronezh, 1982 (in Russian).

    Google Scholar 

  19. Gliklikh Yu.E. and Fedorenko I.V. On geometrization of one class of mechanical systems with random force perturbations. Deposited at VINITI, October 21, 1980, No. 4481-80 (in Russian).

    Google Scholar 

  20. Gliklikh Yu.E. and Fedorenko I.V. On equations of geometric mechanics with random force fields, in: Approximate methods of analysing differential equations and their applications. Kuibyshev, 1981, No. 7 (in Russian).

    Google Scholar 

  21. Godbillon C. Géometrie différentielle at mécanique analytique. Hermann, Paris, 1969.

    MATH  Google Scholar 

  22. Gromoll D., Klingenberg W., and Meyer W. Riemannsche Geometrie im grossen. Springer-Verlag, Berlin-Heidelberg-N.Y., 1968.

    Book  MATH  Google Scholar 

  23. Klein F. Vorlesungen über höhere Geometrie (dritte Auflage bearbeitet und herausgegaben von W. Blaschke). Julius Springer, Berlin, 1926.

    Book  Google Scholar 

  24. Lang S. Introduction to differential manifolds. Interscience, N.Y.-London, 1962.

    MATH  Google Scholar 

  25. McKean, Jr. H.P. Stochastic integrals.-Acad. Press, N.Y.-London, 1969.

    MATH  Google Scholar 

  26. Myshkis A.D. On certain problems in the theory of differential equations with deviated argument.-Usp. Matem. Nauk, 1977, 32, No. 2 (in Russian).

    Google Scholar 

  27. Synge J.L. Hodographs of general dynamical systems.-Trans. Roy. Soc. Canada, Ser. 3, 1926, vol. 25.

    Google Scholar 

  28. Synge J.L. Tensorial methods in dynamics. University of Toronto, 1936.

    Google Scholar 

  29. Sternberg S. Lectures on differential geometry. Prentice Hall, N.J., 1964.

    MATH  Google Scholar 

  30. Filippov A.F. Differential equations with discontinuous right-hand side.-Matem. sbornik, 1960, 51, No. 1 (in Russian).

    Google Scholar 

  31. Hartman P. Ordinary differential equations.-Wiley, N.Y.-London-Sydney, 1964.

    MATH  Google Scholar 

  32. Schmutzer E. Grundprinzipien der klassischen Mechanik und der klassischen Feldtheorie. VEB Deutscher Verlag der Wissenschaften, Berlin, 1973.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Yurii G. Borisovich Yurii E. Gliklikh A. M. Vershik

Rights and permissions

Reprints and permissions

Copyright information

© 1984 Springer-Verlag

About this paper

Cite this paper

Gliklikh, Y.E. (1984). Riemannian parallel translation in non-linear mechanics. In: Borisovich, Y.G., Gliklikh, Y.E., Vershik, A.M. (eds) Global Analysis — Studies and Applications I. Lecture Notes in Mathematics, vol 1108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099555

Download citation

  • DOI: https://doi.org/10.1007/BFb0099555

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13910-2

  • Online ISBN: 978-3-540-39132-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics