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Some Applications of Differential Geometry in Mechanics

  • Kurt Schlacher
  • Gernot Grabmair
  • Helmut Ennsbrunner
  • Richard Stadlmayr
Part of the International Centre for Mechanical Sciences book series (CISM, volume 444)

Abstract

Classical mechanics is not only one of the most successful scientific disciplines, it stands also at the beginning of modern physics. Furthermore, it demonstrates the deep connection between physics and geometry. Therefore, the development of differential geometry was pushed by ideas from mechanics. The goal of this contribution is to show the geometric interpretation of certain mechanical ideas like the time-space manifold or the metric on the spatial manifold and geodesics. The covariant derivative, derived from a special connection, allows us to represent the famous equations of point mechanics in a coordinate free manner. Based on this approach we extend these geometric ideas such that we are able to discuss concepts required for continuum mechanics. Here new geometric concepts like the covariant differential are introduced, which allow to transfer several ideas from point mechanics to continuum mechanics.

Keywords

Vector Bundle Covariant Derivative Linear Momentum Cauchy Stress Tensor Linear Connection 
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.

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Copyright information

© Springer-Verlag Wien 2004

Authors and Affiliations

  • Kurt Schlacher
    • 1
  • Gernot Grabmair
    • 1
  • Helmut Ennsbrunner
    • 1
  • Richard Stadlmayr
    • 1
  1. 1.Institute of Automatic Control and Control Systems TechnologyJohannes Kepler University of LinzAustria

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