Applications of contact geometry and topology in physics - Kholodenko, Arkady Leonidovich (World Scientific, 2013)
Includes bibliographical references and index
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