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Geometric Dynamics

  • Book
  • © 2000

Overview

Authors:
  1. Constantin Udrişte

    1. Department of Mathematics and Physics, University Politehnica of Bucharest, Bucharest, Romania

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Part of the book series: Mathematics and Its Applications (MAIA, volume 513)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Vector Fields

    • Constantin Udrişte
    Pages 1-33

  3. Particular Vector Fields

    • Constantin Udrişte
    Pages 35-61

  4. Field Lines

    • Constantin Udrişte
    Pages 63-116

  5. Stability of Equilibrium Points

    • Constantin Udrişte
    Pages 117-144

  6. Potential Differential Systems of Order One and Catastrophe Theory

    • Constantin Udrişte
    Pages 145-176

  7. Field Hypersurfaces

    • Constantin Udrişte
    Pages 177-200

  8. Bifurcation Theory

    • Constantin Udrişte
    Pages 201-223

  9. Submanifolds Orthogonal to Field Lines

    • Constantin Udrişte
    Pages 225-272

  10. Dynamics Induced by a Vector Field

    • Constantin Udrişte
    Pages 273-302

  11. Magnetic Dynamical Systems and Sabba Ştefănescu Conjectures

    • Constantin Udrişte
    Pages 303-355

  12. Bifurcations in the Mechanics of Hypoelastic Granular Materials

    • Lucia Drăguşin
    Pages 357-384

  13. Back Matter

    Pages 385-395
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Keywords

About this book

Geometric dynamics is a tool for developing a mathematical representation of real world phenomena, based on the notion of a field line described in two ways: -as the solution of any Cauchy problem associated to a first-order autonomous differential system; -as the solution of a certain Cauchy problem associated to a second-order conservative prolongation of the initial system. The basic novelty of our book is the discovery that a field line is a geodesic of a suitable geometrical structure on a given space (Lorentz-Udri~te world-force law). In other words, we create a wider class of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds, ensuring that all trajectories of a given vector field are geodesics. This is our contribution to an old open problem studied by H. Poincare, S. Sasaki and others. From the kinematic viewpoint of corpuscular intuition, a field line shows the trajectory followed by a particle at a point of the definition domain of a vector field, if the particle is sensitive to the related type of field. Therefore, field lines appear in a natural way in problems of theoretical mechanics, fluid mechanics, physics, thermodynamics, biology, chemistry, etc.

Authors and Affiliations

  • Department of Mathematics and Physics, University Politehnica of Bucharest, Bucharest, Romania

    Constantin Udrişte

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