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Field name	Details
Dewey Class	003.3
Title	Geometric Dynamics ([EBook] /) / by Constantin Udrişte.
Author	Udrişte, Constantin
Other name(s)	SpringerLink (Online service)
Publication	Dordrecht : : Springer Netherlands : : Imprint: Springer, , 2000.
Physical Details	XVI, 395 p. : online resource.
Series	Mathematics and its applications ; 513
ISBN	9789401141871
Summary Note	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.:
Contents note	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 of swallowtail -- 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 forming vector 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.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site	http://dx.doi.org/10.1007/978-94-011-4187-1
Links to Related Works	Subject References: Applications of Mathematics . Applied mathematics . Computational Mathematics and Numerical Analysis . Computer mathematics . Differential Geometry . Engineering mathematics . Mathematical Modeling and Industrial Mathematics . Mathematical models . Mathematical optimization . Mathematics . Optimization . Authors: Udrişte, Constantin . Corporate Authors: SpringerLink (Online service) . Series: Mathematics and its applications . Classification: 003.3 .

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