Klassik mexanikada **Hannay burchagi** aylanma geometrik fazaning (yoki Berry fazasining) mexanik analogidir. U Buyuk Britaniyaning Bristol universitetidan Jon Hannay sharafiga nomlangan. Xannay birinchi marta 1985 yilda burchakni tasvirlab, yaqinda rasmiylashtirilgan Berry bosqichi gʻoyalarini klassik mexanikaga kengaytirdi.

Klassik mexanikada Hannay burchagi
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Hannay burchagi harakat-burchak koordinatalari kontekstida aniqlanadi. Dastlab vaqt oʻzgarmaydigan tizimda harakat oʻzgaruvchisi {\displaystyle I\_{\alpha }} doimiy hisoblanadi. Vaqti-vaqti bilan bezovtalanishni kiritgandan soʻng {\displaystyle \lambda (t)}, harakat oʻzgaruvchisi {\displaystyle I\_{\alpha }} adiabatik invariantga aylanadi va Hannay burchagi {\displaystyle \theta \_{\alpha }^{H}} uning mos burchak oʻzgaruvchisi uchun buzilish sodir boʻlgan evolyutsiyani ifodalovchi yoʻl integraliga koʻra hisoblanishi mumkin. {\displaystyle \lambda (t)} asl qiymatiga qaytadi:


θ

α

H

=
−

∂

∂

I

α


∮

p

⋅

∂

q

∂
λ

d

λ

{\displaystyle \theta \_{\alpha }^{H}=-{\frac {\partial }{\partial I\_{\alpha }}}\oint \!{\boldsymbol {p}}\cdot {\frac {\partial {\boldsymbol {q}}}{\partial \lambda }}\mathrm {d} \lambda }

{\displaystyle \theta _{\alpha }^{H}=-{\frac {\partial }{\partial I_{\alpha }}}\oint \!{\boldsymbol {p}}\cdot {\frac {\partial {\boldsymbol {q}}}{\partial \lambda }}\mathrm {d} \lambda }bu yerda {\displaystyle {\boldsymbol {p}}} va {\displaystyle {\boldsymbol {q}}} Gamiltonianning kanonik oʻzgaruvchilari .

Misol
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Fuko mayatnik klassik mexanikaning namunasidir, u baʼzan Berri fazasini tasvirlash uchun ham ishlatiladi. Quyida biz harakat burchagi oʻzgaruvchilari yordamida Fuko mayatnikini oʻrganamiz. Oddiylik uchun biz umumiy protokolda qoʻllaniladigan Gamilton-Jakobi tenglamasidan foydalanmaymiz.

Biz chastotali tekislik mayatnikini koʻrib chiqamiz {\displaystyle \omega } burchak tezligi boʻlgan Yerning aylanishi taʼsirida {\displaystyle {\vec {\Omega }}=(\Omega \_{x},\Omega \_{y},\Omega \_{z})} sifatida belgilangan amplituda bilan {\displaystyle \Omega =|{\vec {\Omega }}|}. Mana, {\displaystyle z} yoʻnalishi Yerning markazidan mayatnikgacha. Mayatnik uchun Lagrangian:



L
=

1
2

m
(

x
˙

2

+

y
˙

2

)
−

1
2

m

ω

2

(

x

2

+

y

2

)
+
m

Ω

z

(
x

y
˙

−
y

x
˙

)

{\displaystyle L={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2})-{\frac {1}{2}}m\omega ^{2}(x^{2}+y^{2})+m\Omega \_{z}(x{\dot {y}}-y{\dot {x}})}

{\displaystyle L={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2})-{\frac {1}{2}}m\omega ^{2}(x^{2}+y^{2})+m\Omega _{z}(x{\dot {y}}-y{\dot {x}})}Tegishli harakat tenglamasi



x
¨

+

ω

2

x
=
2

Ω

z


y
˙


{\displaystyle {\ddot {x}}+\omega ^{2}x=2\Omega \_{z}{\dot {y}}}

{\displaystyle {\ddot {x}}+\omega ^{2}x=2\Omega _{z}{\dot {y}}}



y
¨

+

ω

2

y
=
−
2

Ω

z


x
˙


{\displaystyle {\ddot {y}}+\omega ^{2}y=-2\Omega \_{z}{\dot {x}}}

{\displaystyle {\ddot {y}}+\omega ^{2}y=-2\Omega _{z}{\dot {x}}}Keyin yordamchi oʻzgaruvchini kiritamiz {\displaystyle \varpi =x+iy} bu aslida burchak oʻzgaruvchisi. Endi biz uchun tenglama mavjud {\displaystyle \varpi } :



ϖ
¨

+

ω

2

ϖ
=
−
2
i

Ω

z


ϖ
˙


{\displaystyle {\ddot {\varpi }}+\omega ^{2}\varpi =-2i\Omega \_{z}{\dot {\varpi }}}

{\displaystyle {\ddot {\varpi }}+\omega ^{2}\varpi =-2i\Omega _{z}{\dot {\varpi }}}Uning xarakteristik tenglamasidan



λ

2

+

ω

2

=
−
2
i

Ω

z

λ

{\displaystyle \lambda ^{2}+\omega ^{2}=-2i\Omega \_{z}\lambda }

{\displaystyle \lambda ^{2}+\omega ^{2}=-2i\Omega _{z}\lambda }biz uning xarakterli ildizini olamiz (biz shuni taʼkidlaymiz {\displaystyle \Omega \ll \omega })


λ
=
−
i

Ω

z

±
i

Ω

z

2

+

ω

2

≈
−
i

Ω

z

±
i
ω

{\displaystyle \lambda =-i\Omega \_{z}\pm i{\sqrt {\Omega \_{z}^{2}+\omega ^{2}}}\approx -i\Omega \_{z}\pm i\omega }

{\displaystyle \lambda =-i\Omega _{z}\pm i{\sqrt {\Omega _{z}^{2}+\omega ^{2}}}\approx -i\Omega _{z}\pm i\omega }Yechim shundan keyin


ϖ
=

e

−
i

Ω

z

t

(
A

e

i
ω
t

+
B

e

−
i
ω
t

)

{\displaystyle \varpi =e^{-i\Omega \_{z}t}(Ae^{i\omega t}+Be^{-i\omega t})}

{\displaystyle \varpi =e^{-i\Omega _{z}t}(Ae^{i\omega t}+Be^{-i\omega t})}Yer bir toʻliq aylanishdan keyin, yaʼni {\displaystyle T=2\pi /\Omega \approx 24h}, biz uchun faza oʻzgarishi bor {\displaystyle \varpi }


Δ
φ
=
2
π

ω
Ω

+
2
π

Ω

z

Ω

{\displaystyle \Delta \varphi =2\pi {\frac {\omega }{\Omega }}+2\pi {\frac {\Omega \_{z}}{\Omega }}}

{\displaystyle \Delta \varphi =2\pi {\frac {\omega }{\Omega }}+2\pi {\frac {\Omega _{z}}{\Omega }}}Birinchi atama mayatnikning dinamik taʼsiriga bogʻliq boʻlib, dinamik faza deb ataladi, ikkinchisi esa Hannay burchagi boʻlgan geometrik fazani ifodalaydi.


θ

H

=
2
π

Ω

z

Ω

{\displaystyle \theta ^{H}=2\pi {\frac {\Omega \_{z}}{\Omega }}}

{\displaystyle \theta ^{H}=2\pi {\frac {\Omega _{z}}{\Omega }}}

Manbalar
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1. ↑  Toshikaze Kariyado; Yasuhiro Hatsugai (2016). "Hannay Angle: Yet Another Symmetry-Protected Topological Order Parameter in Classical Mechanics". *J. Phys. Soc. Jpn.* **85** (4): 043001. doi:10.7566/JPSJ.85.043001.

* Hannay John H. 1985 Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian J. Phys. A: Math. Gen. **18** 221-30.
* Marsden, Jerrold E.; Montgomery, Richard; Ratiu, Tudor S.*. Reduction, Symmetry, and Phases in Mechanics*. AMS Bookstore, 1990 — 69 bet. ISBN 0-8218-2498-8.
* C. Pisani*. Quantum-mechanical Ab-initio Calculation of the Properties of Crystalline Materials*, Proceedings of the IV School of Computational Chemistry of the Italian Chemical Society, Springer, 1994 — 282 bet. ISBN 3-540-61645-4.
* ; Jean-Marc Triscone; Charles H Ahn*Physics of Ferroelectrics: a Modern Perspective*. Springer, 2007 — 2 bet. ISBN 978-3-540-34590-9.

Havolalar
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* Professor John H. Hannay: Research Highlights. Department of Physics, University of Bristol.