The Resource Applications of contact geometry and topology in physics, by Arkady L Kholodenko (Clemson University, USA)

Applications of contact geometry and topology in physics, by Arkady L Kholodenko (Clemson University, USA)

Applications of contact geometry and topology in physics
Applications of contact geometry and topology in physics
Statement of responsibility
by Arkady L Kholodenko (Clemson University, USA)
Although contact geometry and topology is briefly discussed in V I Arnold's book "Mathematical Methods of Classical Mechanics" (Springer-Verlag, 1989, 2nd edition), it still remains a domain of research in pure mathematics, e.g. see the recent monograph by H Geiges "An Introduction to Contact Topology" (Cambridge U Press, 2008). Some attempts to use contact geometry in physics were made in the monograph "Contact Geometry and Nonlinear Differential Equations" (Cambridge U Press, 2007). Unfortunately, even the excellent style of this monograph is not sufficient to attract the attention of the physics community to this type of problems. This book is the first serious attempt to change the existing status quo. In it we demonstrate that, in fact, all branches of theoretical physics can be rewritten in the language of contact geometry and topology: from mechanics, thermodynamics and electrodynamics to optics, gauge fields and gravity; from physics of liquid crystals to quantum mechanics and quantum computers, etc. The book is written in the style of famous Landau-Lifshitz (L-L) multivolume course in theoretical physics. This means that its readers are expected to have solid background in theoretical physics (at least at the level of the L-L course). No prior knowledge of specialized mathematics is required. All needed new mathematics is given in the context of discussed physical problems. As in the L-L course some problems/exercises are formulated along the way and, again as in the L-L course, these are always supplemented by either solutions or by hints (with exact references). Unlike the L-L course, though, some definitions, theorems, and remarks are also presented. This is done with the purpose of stimulating the interest of our readers in deeper study of subject matters discussed in the text
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index present
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K46 2013eb
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non fiction
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  • dictionaries
  • bibliography
Applications of contact geometry and topology in physics, by Arkady L Kholodenko (Clemson University, USA)
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Bibliography note
Includes bibliographical references and index
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online resource
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Ch. 1. Motivation and background. 1.1. General information. 1.2. Fluid mechanics formulation of Hamiltonian and Jacobian mechanics. Emergence of the force-free fields. 1.3. Some basic facts about the force-free fields -- ch. 2. From ideal magnetohydrodynamics to string and knot theory. 2.1. General information. 2.2. The Gillbarg problem and the theory of foliations. 2.3. From string-theoretic Lund-Regge equation to Landau-Lifshitz equation for the vortex filament. 2.4. Foliations of R[symbol] by the Maxwellian surfaces. 2.5. The Maxwellian tori and the torus knots associated with them -- ch. 3. All about and around Woltjer's theorem. 3.1. General information. 3.2. Equilibria in liquid crystals and the Faddeev-Skyrme model for pure Yang-Mills fields. 3.3. Refinements of Woltjer's theorem. Implications for magnetohydrodynamics, superconductivity and liquid crystals. 3.4. Proca's massive electrodynamics and Stueckelberg's trick. 3.5. New interpretation of the Dirac monopole and its use in the problem of quark confinement -- ch. 4. Topologically massive gauge theories and the force-free fields -- ch. 5. Contact geometry and physics. 5.1. General information. 5.2. Some basic facts about contact geometry and topology. 5.3. Contact geometry of thermodynamics. 5.4. Contact and symplectic geometry and liquid crystals. 5.5. Force-free (Beltrami) fields and contact geometry and topology of hydrodynamics and electromagnetism. 5.6. Many facets of the Abelian Chern-Simons functional and their relation to monopoles, dyons and the Faddeev-Skyrme model -- ch. 6. Sub-Riemannian geometry, Heisenberg manifolds and quantum mechanics of Landau levels. 6.1. Motivation. 6.2. The benchmark example. 6.3. Basics of sub-Riemannian geometry. 6.4. Glimpses of quantum mechanics. 6.5. Fiber bundle reformulation of sub-Riemannian geometry and classical-quantum correspondence. Connection with Dirac monopoles -- ch. 7. Abrikosov lattices, TGB phases in liquid crystals and Heisenberg group -- ch. 8. Sub-Riemannian geometry, spin dynamics and quantum-classical optimal control. 8.1. General information. 8.2. Quantum computers paradigm and dynamics of 2-level quantum systems. 8.3. Beyond the 2-level quantum systems. 8.4. Semiflexible polymers and quantum computers -- ch. 9. From contact geometry to contact topology. 9.1. General information. 9.2. Mathematics and physics of the Cauchy problem in quantum mechanics: Viktor Maslov versus David Bohm. 9.3. From Maslov and Bohm to Bell and beyond. 9.4. Harmonious coexistence of classical and quantum mechanics: all about and around the Duistermaat-Heckman formula. 9.5. Mathematics and physics of Weinstein's conjecture: from classical statistical mechanics to Seiberg-Witten monopoles. 9.6. Quantum money, Lagrangian, Legendrian and transverse knots and links and the associated grid diagrams. 9.7. Latest developments in contact geometry and topology. A guided tour with physics applications in the perspective
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  • (OCoLC)847526798
  • acaebk9814412090

Library Locations

    • Deakin University Library - Geelong Waurn Ponds CampusBorrow it
      75 Pigdons Road, Waurn Ponds, Victoria, 3216, AU
      -38.195656 144.304955
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