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

By Arkady L Kholodenko

Even though touch geometry and topology is in short mentioned in V I Arnol'd's booklet "Mathematical equipment of Classical Mechanics "(Springer-Verlag, 1989, 2d edition), it nonetheless continues to be a website of analysis in natural arithmetic, e.g. see the hot monograph via H Geiges "An advent to touch Topology" (Cambridge U Press, 2008). a few makes an attempt to take advantage of 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 enough to draw the eye of the physics group to this sort of difficulties. This booklet is the 1st critical try and swap the prevailing establishment. In it we display that, in reality, all branches of theoretical physics will 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 pcs, and so forth. The booklet is written within the type of well-known Landau-Lifshitz (L-L) multivolume direction in theoretical physics. which means its readers are anticipated to have reliable heritage in theoretical physics (at least on the point of the L-L course). No past wisdom of specialised arithmetic is needed. All wanted new arithmetic is given within the context of mentioned actual difficulties. As within the L-L direction a few problems/exercises are formulated alongside the way in which and, back as within the L-L direction, those are continuously supplemented by way of both suggestions or by means of tricks (with detailed references). in contrast to the L-L direction, although, a few definitions, theorems, and feedback also are offered. this can be performed with the aim of stimulating the curiosity of our readers in deeper examine of topics mentioned within the textual content.

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

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**Applications of Contact Geometry and Topology in Physics**

Even supposing touch geometry and topology is in brief mentioned in V I Arnol'd's ebook "Mathematical tools 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 through H Geiges "An creation to touch Topology" (Cambridge U Press, 2008).

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The simplest knot with considerable symmetry, the ﬁgure eight knot, made of piece of springy wire may have selfintersection points. Langer and Singer conjectured that there should be at least three such points (counting multiplicity) in a knotted wire. 2. Recall that the knot K is a torus knot if it can be embedded into the surface of a torus [36]. 3. Based on the above facts it remains to connect solutions of the L-L Eq. 1. In order to place such solutions onto the Maxwellian tori. The above self-intersection conjecture has been analyzed further in a number of papers.

5) with ±κα replaced by −δ −2 . That is we re-obtain Eq. 3) in which −α2 is replaced by −δ −2 . The functional, Eq. 3), can be looked upon as time-independent version of the so-called Proca functional describing the massive Abelian gauge vector ﬁeld [95]. 4) FG-L [ψ, A] = where the tensor F µν = ∂ µ Aν − ∂ ν Aµ , and its dual F˜ µν is ∗F µν = 12 εµνρσ Fρσ . j ν is the four-dimensional current j = (ρ, j). 5) March 19, 2013 10:56 9in x 6in Applications of Contact Geometry and Topology in Physics All About and Around Woltjer’s Theorem b1524-ch03 41 where is the d’Alambertian, that is = ∂t2 − ∇2 .

B1524-ch03 29 (b) Two types of Hopf links: (a) the right-hand Hopf link, (b) the left-hand Hopf link. The thickening of tubes can continue only till they reach the conﬁguration when the cross-section of one tube will be tightly bounded by another torus tube and vice versa. This conﬁguration bounds the energy from below. These arguments bring us to the necessity of introducing the volumes Vi of tubes for each link component. Without loss of generality, it is then possible to focus attention only at one knotted torus.