Ebook: Geometric Dynamics
Author: Constantin Udrişte (auth.)
- Tags: Mathematical Modeling and Industrial Mathematics, Applications of Mathematics, Differential Geometry, Optimization, Computational Mathematics and Numerical Analysis
- Series: Mathematics and Its Applications 513
- Year: 2000
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- 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.
Content:
Front Matter....Pages i-xvi
Vector Fields....Pages 1-33
Particular Vector Fields....Pages 35-61
Field Lines....Pages 63-116
Stability of Equilibrium Points....Pages 117-144
Potential Differential Systems of Order One and Catastrophe Theory....Pages 145-176
Field Hypersurfaces....Pages 177-200
Bifurcation Theory....Pages 201-223
Submanifolds Orthogonal to Field Lines....Pages 225-272
Dynamics Induced by a Vector Field....Pages 273-302
Magnetic Dynamical Systems and Sabba ?tef?nescu Conjectures....Pages 303-355
Bifurcations in the Mechanics of Hypoelastic Granular Materials....Pages 357-384
Back Matter....Pages 385-395
Content:
Front Matter....Pages i-xvi
Vector Fields....Pages 1-33
Particular Vector Fields....Pages 35-61
Field Lines....Pages 63-116
Stability of Equilibrium Points....Pages 117-144
Potential Differential Systems of Order One and Catastrophe Theory....Pages 145-176
Field Hypersurfaces....Pages 177-200
Bifurcation Theory....Pages 201-223
Submanifolds Orthogonal to Field Lines....Pages 225-272
Dynamics Induced by a Vector Field....Pages 273-302
Magnetic Dynamical Systems and Sabba ?tef?nescu Conjectures....Pages 303-355
Bifurcations in the Mechanics of Hypoelastic Granular Materials....Pages 357-384
Back Matter....Pages 385-395
....