Acta Applicandae Mathematica

, Volume 70, Issue 1–3, pp 209–230

Pseudo-Rigid Bodies: A Geometric Lagrangian Approach

  • M. Esmeralda Sousa Dias
Article

Abstract

The pseudo-rigid body model is viewed within the context of continuum mechanics and elasticity theory. A Lagrangian reduction, based on variational principles, is developed for both anisotropic and isotropic pseudo-rigid bodies. For isotropic Lagrangians, the reduced equations of motion for the pseudo-rigid body are a system of two (coupled) Lax equations on so(3)×so(3) and a second-order differential equation on the set of diagonal matrices with a positive determinant. Several examples of pseudo-rigid bodies such as stretching bodies, spinning gas cloud and Riemann ellipsoids are presented.

symmetry reduction pseudo-rigid bodies Euler–Lagrange equations isotropic anisotropic 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cendra, H., Marsden, J. E. and Ratiu, T. S.: Lagrangian reduction by stages, Mem. Amer. Math. Soc. 152 (2001); [http://www.cds.caltech.edu/~marsden/].Google Scholar
  2. 2.
    Cendra, H., Marsden, J. E. and Ratiu, T. S.: Geometric mechanics, Lagrangian reduction, and nonholonomic systems, In: Mathematics Unlimited-2001 and Beyond, Springer, New York, 2001.Google Scholar
  3. 3.
    Chandrasekhar, S.: Ellipsoidal Figures of Equilibrium, Dover, New York, 1987.Google Scholar
  4. 4.
    Ciarlet, P. G.: Mathematical Elasticity, vol. 1: Three-dimensional Elasticity, North-Holland, Amsterdam, 1993.Google Scholar
  5. 5.
    Cohen, H. and Muncaster, R. G.: The Theory of Pseudo-rigid Bodies, Tracts in Natural Philos. 33, Springer, New York, 1988.Google Scholar
  6. 6.
    Dyson, F. J.: Dynamics of a spinning gas cloud, J. Math. Mech. 18(1) (1968), 91-101.Google Scholar
  7. 7.
    Lewis, D. and Simo, J. C.: Nonlinear stability of rotating pseudo-rigid bodies, Proc. Royal Soc. London A 427 (1990), 281-319.Google Scholar
  8. 8.
    Gurtin, M. E.: An introduction to Continuum Mechanics, Math. Sci. Engrg. 138, Academic Press, New York, 1981.Google Scholar
  9. 9.
    Kato, T.: Perturbation Theory of Linear Operators, Springer, New York, 1980.Google Scholar
  10. 10.
    Marsden, J. E. and Hughes, T. J. R.: Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, 1983.Google Scholar
  11. 11.
    Marsden, J. E. and Ratiu, T. S.: Introduction to Mechanics and Symmetry, 2nd edn, Texts in Appl. Math. 75, Springer, New York, 1999.Google Scholar
  12. 12.
    Roberts, R. M. and Sousa-Dias, M. E.: Symmetries of Riemann ellipsoids, Resenhas IME-USP 4(2) (1999), 183-221.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • M. Esmeralda Sousa Dias
    • 1
  1. 1.Dep. MatemáticaInstituto Superior TécnicoLisboaPortugal

Personalised recommendations