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…mehr
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.
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.
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.
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