Applications of Contact Geometry and Topology in Physics by Arkady L Kholodenko

By Arkady L Kholodenko

Even though touch geometry and topology is in brief mentioned in V I Arnol'd's e-book "Mathematical equipment of Classical Mechanics "(Springer-Verlag, 1989, second edition), it nonetheless is still a site of study in natural arithmetic, e.g. see the hot monograph by way of H Geiges "An advent to touch Topology" (Cambridge U Press, 2008). a few makes an attempt to exploit touch geometry in physics have been made within the monograph "Contact Geometry and Nonlinear Differential Equations" (Cambridge U Press, 2007). regrettably, even the superb variety of this monograph isn't really adequate to draw the eye of the physics group to this kind of difficulties. This publication is the 1st critical try to swap the present establishment. In it we exhibit that, in truth, all branches of theoretical physics might be rewritten within the language of touch geometry and topology: from mechanics, thermodynamics and electrodynamics to optics, gauge fields and gravity; from physics of liquid crystals to quantum mechanics and quantum desktops, and so forth. The publication is written within the type of recognized Landau-Lifshitz (L-L) multivolume direction in theoretical physics. which means its readers are anticipated to have reliable history in theoretical physics (at least on the point of the L-L course). No previous wisdom of specialised arithmetic is needed. All wanted new arithmetic is given within the context of mentioned actual difficulties. As within the L-L path a few problems/exercises are formulated alongside the way in which and, back as within the L-L direction, those are continually supplemented via both options or by means of tricks (with particular references). not like the L-L path, notwithstanding, a few definitions, theorems, and feedback also are offered. this is often performed with the aim of stimulating the curiosity of our readers in deeper learn of topics mentioned within the textual content.

Readership: Researchers and execs in utilized arithmetic and theoretical physics.

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Extra resources for Applications of Contact Geometry and Topology in Physics

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1. However, Kida considered not only closed filaments forming torus knots but also open filaments. There is no need for us to discuss such configurations as results of the next chapter indicate. In the meantime, we would like to mention the following. In [53] Ricca reobtained Kida’s results for torus knots by much simpler methods. Refinements of his results were made by Calini and Ivey [54–58]. A very detailed book by Kambe [59] also provides the wealth of useful related information. All these authors use essentially the Hasimoto transform.

2. 1. Investigate if such an equation is obtainable from the theory of elastic rods. Hint: Look at [16–18] mentioned in [33]. Thus, Theorem 1 of Rogers and Schiff, [33], indeed describes the superconducting-type solution for plasma as well as for smectics A in accord March 19, 2013 10:56 9in x 6in Applications of Contact Geometry and Topology in Physics b1524-ch03 Applications of Contact Geometry and Topology in Physics 40 with de Gennes [77] whose derivation of this result is entirely different.

There are two ways to restore the gauge invariance. One is to consider the so-called Higgs mechanism of mass generation which in the present case boils down to using the full G-L functional, Eq. 7), which is manifestly gauge invariant. Another is to consider the so-called Stuekelberg trick [96]. Instead of Proca’s Lagrangian, Eq. 8), Stueckelberg suggested to use 1 † µν 1 F + m2 Vµ† − ∂µ B † LStueck = − Fµν 2 m Vµ− 1 ∂µ B m − (∂µ V †µ + mB † )(∂µ V µ + mB). 10) If for a moment we would ignore the last term in this expression, in the static case we would re-obtain back the London approximation to the G-L functional, that is Eq.

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