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.
Reihe
Auflage
Softcover reprint of the original 1st ed. 2000
Sprache
Verlagsort
Zielgruppe
Für Beruf und Forschung
Research
Illustrationen
Maße
Höhe: 240 mm
Breite: 160 mm
Dicke: 23 mm
Gewicht
ISBN-13
978-94-010-5822-3 (9789401058223)
DOI
10.1007/978-94-011-4187-1
Schweitzer Klassifikation
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.- 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.- 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.- 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.- 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.
