Geometric Dynamics

1. Front Matter

   Pages i-xvi

   PDF

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

    PDF

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.

• dynamics
• geometry
• manifold
• mathematics
• mechanics

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

  Constantin Udrişte

Policies and ethics