Geometric Dynamics

ISBN-13: 9789401058223 / Angielski / Miękka / 2012 / 395 str.

Constantin Udriste

Opis

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.

Spis treści

Preface. 1. Vector Fields. 2. Particular Vector Fields. 3. Field Lines. 4. Stability of Equilibrium Points. 5. Potential Differential Systems of Order One and Catastrophe Theory. 6. Field Hypersurfaces. 7. Bifurcation Theory. 8. Submanifolds Orthogonal to Field Lines. 9. Dynamics Induced by a Vector Field. 10. Magnetic Dynamical Systems and Sabba Stefanescu Conjectures. 11. Bifurcations in the Mechanics of Hypoelastic Granular Materials; L. Dragusin. Bibliography. Index.