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Geometric Dynamics

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 proble...

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Bibliographic Details
Main Author: Udriste, C.
Format: eBook
Language:English
Published: Dordrecht Springer Netherlands 2000, 2000
Edition:1st ed. 2000
Series:Mathematics and Its Applications
Subjects:
Geometry, Differential
Optimization
Computational Mathematics And Numerical Analysis
Mathematics / Data Processing
Mathematical Modeling And Industrial Mathematics
Differential Geometry
Applications Of Mathematics
Mathematics
Mathematical Optimization
Mathematical Models
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 3.9. Local flow generated by a conformai vector field
  • 3.10. Local flow generated by a projective vector field
  • 3.11. Local flow generated by an irrotational, solenoidal or torse forming vector field
  • 3.12. Vector fields attached to the local groups of diffeomorphisms
  • 3.13. Proposed problems
  • 4 Stability of Equilibrium Points
  • 4.1. Problem of stability
  • 4.2. Stability of zeros of linear vector fields
  • 4.3. Classification of equilibrium points in the plane
  • 4.4. Stability by linear approximation
  • 4.5. Stability by Lyapunov functions
  • 4.6. Proposed problems
  • 5. Potential Differential Systems of Order One and Catastrophe Theory
  • 5.1. Critical points and gradient lines
  • 5.2. Potential differential systems and elementary catastrophes
  • 5.3. Gradient lines of the fold
  • 5.4. Gradient lines of the cusp
  • 5.5. Equilibrium points of gradient ofswallowtail
  • 5.6. Equilibrium points of gradient of butterfly
  • 1 Vector Fields
  • 1.1. Scalar fields
  • 1.2. Vector fields
  • 1.3. Submanifolds of Rn
  • 1.4. Derivative with respect to a vector
  • 1.5. Vector fields as linear operators and derivations
  • 1.6. Differential operators
  • 1.7. Proposed problems
  • 2 Particular Vector Fields
  • 2.1. Irrotational vector fields
  • 2.2. Vector fields with spherical symmetry
  • 2.3. Solenoidal vector fields
  • 2.4. Monge and Stokes representations
  • 2.5. Harmonic vector fields
  • 2.6. Killing vector fields
  • 2.7. Conformai vector fields
  • 2.8. Affine and projective vector fields
  • 2.9. Torse forming vector fields
  • 2.10. Proposed problems
  • 3 Field Lines
  • 3.1. Field lines
  • 3.2. First integrals
  • 3.3. Field lines of linear vector fields
  • 3.4. Runge-Kutta method
  • 3.5. Completeness of vector fields
  • 3.6. Completeness of Hamiltonian vector fields
  • 3.7. Flows and Liouville’s theorem
  • 3.8. Global flow generated by a Killing or affine vector field
  • 5.7. Equilibrium points of gradient of elliptic umbilic
  • 5.8. Equilibrium points of gradient of hyperbolic umbilic
  • 5.9. Equilibrium points of gradient of parabolic umbilic
  • 5.10. Proposed problems
  • 6. Field Hypersurfaces
  • 6.1. Linear equations with partial derivatives of first order
  • 6.2. Homogeneous functions and Euler’s equation
  • 6.3. Ruled hypersurfaces
  • 6.4. Hypersurfaces of revolution
  • 6.5. Proper values and proper vectors of a vector field
  • 6.6. Grid method
  • 6.7. Proposed problems
  • 7. Bifurcation Theory
  • 7.1 Bifurcation in the equilibrium set
  • 7.2 Centre manifold
  • 7.3 Flow bifurcation
  • 7.4 Hopf theorem of bifurcation
  • 7.5 Proposed problems
  • 8. Submanifolds Orthogonal to Field Lines
  • 8.1. Submanifolds orthogonal to field lines
  • 8.2. Completely integrable Pfaff equations
  • 8.3. Frobenius theorem
  • 8.4. Biscalar vector fields
  • 8.5. Distribution orthogonal to a vector field
  • 8.6. Field lines as intersections of nonholonomic spaces
  • 10.1. Biot-Savart-Laplace dynamical systems
  • 10.2. Sabba ?tef?nescu conjectures
  • 10.3. Magnetic dynamics around filiform electric circuits of right angle type
  • 10.4. Energy of magnetic field generated by filiform electric circuits of right angle type
  • 10.5. Electromagnetic dynamical systems as Hamiltonian systems
  • 11 Bifurcations in the Mechanics of Hypoelastic Granular Materials
  • 11.1. Constitutive Equations
  • 11.2. The Axial Symmetric Case
  • 11.3. Conclusions
  • 11.4. References
  • 8.7. Distribution orthogonal to an affine vector field
  • 8.8. Parameter dependence of submanifolds orthogonal to field lines
  • 8.9. Extrema with nonholonomic constraints
  • 8.10. Thermodynamic systems and their interaction
  • 8.11. Proposed problems
  • 9. Dynamics Induced by a Vector Field
  • 9.1. Energy and flow of a vector field
  • 9.2. Differential equations of motion in Lagrangian and Hamiltonian form
  • 9.3. New geometrical model of particle dynamics
  • 9.4. Dynamics induced by an irrotational vector field
  • 9.5. Dynamics induced by a Killing vector field
  • 9.6. Dynamics induced by a conformai vector field
  • 9.7. Dynamics mduced by an affine vector field
  • 9.8. Dynamics induced by a projective vector field
  • 9.9. Dynamics induced by a torse formingvector field
  • 9.10. Energy of the Hamiltonian vector field
  • 9.11. Kinematic systems of classical thermodynamics
  • 10 Magnetic Dynamical Systems and Sabba ?tef?nescu Conjectures

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