By Arkady L Kholodenko
Even supposing touch geometry and topology is in short mentioned in V I Arnol'd's booklet "Mathematical tools of Classical Mechanics "(Springer-Verlag, 1989, second edition), it nonetheless is still a site of analysis in natural arithmetic, e.g. see the new 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 type of this monograph isn't really adequate to draw the eye of the physics group to this kind of difficulties. This booklet is the 1st critical try to swap the present establishment. In it we display 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 pcs, and so forth. The e-book is written within the form of well-known Landau-Lifshitz (L-L) multivolume direction in theoretical physics. which means its readers are anticipated to have sturdy 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 continually supplemented through both options or by means of tricks (with precise references). not like the L-L direction, even though, 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 themes mentioned within the textual content.
Readership: Researchers and execs in utilized arithmetic and theoretical physics.
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Extra info for Applications of Contact Geometry and Topology in Physics
11a) with lk(1, 2) being the linking number between contours C1 and C2 . Using the last two equations it is possible to rewrite now the energy, Eq. 9), in terms of linking numbers. That is the hydrodynamics of ideal ﬂuids can be reformulated as the Abelian topological ﬁeld theory of the type discussed in . Speciﬁcally, taking into account the non-negativity of E, we obtain: E= k2 |lk(1, 2)|. 11a) coincides with that obtained by Moﬀatt and Ricca , where it was derived using diﬀerent arguments.
2b) in terms of coordinates of the adapted coordinate system. In particular, Eqs. 2a) acquire the following form: div v = div rχ = |rχ , rψ , rω |χ = 0, |rχ , rψ , rω | div rψ = |rχ , rψ , rω |ψ = 0. 1) It is always possible to select the adapted coordinates in such a way that the volume |rχ , rψ , rω | is normalized to one. Under this condition, Eq. 2b) acquires the form rχχ − rψψ + Πχ rψ × rω + Πψ rω × rχ + Πω rχ × rψ = 0. 2) If the Maxwellian surfaces are the constant pressure surfaces, then Π = Π(ω).
See Eq. 8). As a result, we can apply the Arnol’d inequality, Eq. 12), and the rest of results of the previous section. These conclusions are immediately transferable to the F-S model. Indeed, following work by Vakulenko and Kapitanskii  we deﬁne the F-S functional for the static variant of this model given by FF-S [n] ≡ F1 [n] + F2 [n] 2 1 = dr 2 ∂i na ∂i na + 2λ 2 R3 3 (εabc ∂i na ∂k nb nc )2 . 12) i,k=1 As usual, the summation over the repeated indices is assumed. As in the case of liquid crystals, the functional, Eq.