CC BY
10
Vol. 20, No. 02, 2017
Ovil Aviation High Technologies
UDC 519.46
ON THE STRUCTURE OF THE OPERATOR COADJOINT ACTION
FOR THE CURRENT ALGEBRA ON THE THREE-DIMENSIONAL TORUS
A.M. LUKATSKY1
1Energy Research Institute of Russian Academy of Sciences, Moscow, Russia
For the current Lie algebra on the three-dimensional torus with non-standard Lie bracket some properties, in the case when the sum of adjoint and coadjoint operators on infinite-dimensional Lie algebra with scalar product has a finite norm are established. For the Landau-Lifshitz equation in the three-dimensional torus it is established that the operator
Sm = (adm + ad ) / 2 has a finite norm, though it is not true the operators of the adjoint action adm and coadjoint action ad *m . It follows that the coefficients of expansion of the solution in an orthonormal basis of eigenvectors of the Laplace operator satisfy Lipschitz conditions. Thus, for the Landau-Lifshitz equation on the three-dimensional torus situation is similar to the equations of an ideal fluid and Korteweg de Vries. On the other hand, if for the equations of fluid dynamics and Korteweg de Vries, this fact has been established in a general way, for the Landau-Lifshitz equation in the three-dimensional torus it is obtained specifically through the calculation of structural constants and the matrix of the coadjoint action for the current algebra with non-standard Lie bracket.
Key words: current algebra, Lie bracket, operator of the adjoint action, operator of the coadjoint action, three-torus, the Landau-Lifshitz equation, compact operator, Lipschitz condition.
INTRODUCTION
Let us denote the three-dimensional torus by T3. Denote by V(T3) the space of smooth vector fields on T3. In the space V(T3) there exists the standard current algebra structure g(T3,so(3)) , i.e. the Lie algebra of vector fields on T3 with the operation of the pointwise vector product:
(m x n)(x) = m (x)x n (x). (1)
Consider the Landau-Lifshitz equation on T3:
— = m xAm , (2)
dt
Here me V(T3) , A - Laplace operator on vector fields.
We can represent the Landau-Lifshitz equation as Euler equation in the current algebra but with a non-standard Lie bracket [1-4] (see also [6-7] for the invariant approach methods used). For this we fix a > 0 and introduce the operator:
Pa =-aid + A,
where Id is the identity operator. The operator Pa is invertible.
Let there be given in the space V(T3) a non-standard Lie bracket:
ad mn = [m , n] = P-1(Pam x Pan ) • (3)
Научный Вестник МГТУ ГА_Том 20, № 02, 2017
Ovil Aviation High Technologies Vol. 20, No. 02, 2017
Define also the scalar product in V ^ T3 j:
< m, n > — [ 3 (m(x), - Pa (n)(x)) dx. (4)
JT
Denote by ad^n the operator of coadjoint action in the sense of metric (4):
< ad m (u), v >—< u,adm(v) > . (5)
According to [4] the operator of coadjoint action has the form:
adm(n)—- PaadmP- (n) — -Pamxn . (6)
Then the Landau-Lifshitz equation has the form of the Euler equation [4] on the current algebra with Lie bracket (3) and scalar product (4).
Introduce the operator
d-m = ad ! (m ). (7)
ot
Sm = "~(ad m + ad ! ) . (8)
Construct the orthonormal basis of the Lie algebra V(T3) in the sense of the metric (4) using eigenvectors of the operator Pa :
Kk fx г г \---/тл W _ К
< = , k , ¿,2, ) cos(k0, fk = k <Ai, ¿,2, ¿,3) sm^,k ф 0. (9)
V(i k |2 + a) V(| k |2 + a) W
Here k e Z3, i = 1,2,3, К =
1
2
-Гл' k ^ 0; Ко =
vol( T3 ) ^
1 , St 1 = 1 if i = j, otherwise 0.
vol ( T3 )' " ,j
For uniqueness it is convenient to represent an integer vector k ^ 0 (that is, the index of an element of basis (9)) in the form k=e(k)k, where the first non-zero coordinate of the vector k is positive and e(k)e{1,-1}. We use notation [Z3]={k | k e Z3 \ {0}}=Z3 \{0} / {±1}.
Let m(t) be a solution of (7). Expand it to the orthonormal basis (9):
m(t)=£ go, (t) 4 + S (Sk,i (t) ek + hKi (tf ). (10)
i ke[Z3], i
According to [4], the Landau-Lifshitz equation implies the following equation for the coefficients of expansion (10):
ôg dh
-- = -< S i m, m >,-- = -< S^m, m > . (11)
ôt ei ôt ft
Vol. 20, No. 02, 2017
Oivil Aviation High Technologies
In [5] a class of infinite-dimensional Lie with scalar product has been introduced, for which the operator Sm has a finite norm. This property has been established for the Lie algebra of divergence-free vector fields
on a compact Riemannian manifold (the configuration space of an ideal incompressible fluid); Lie algebra of vector fields on a compact Riemannian manifold (the configuration space of an ideal compressible fluid); Virasoro algebra (the configuration space of the Korteweg-de Vries equation). It follows that for the Euler solutions' expansion coefficients in the orthonormal basis of the Lie algebra satisfy Lipschitz conditions.
In this paper, this feature is established for the Landau-Lifshitz equation in the three-torus.
1. THE STRUCTURE PROPERTIES OF THE CURRENT ALGEBRA V(T3) WITH A NON-STANDARD LIE BRACKET
To calculate the structure constants of Lie algebra V(T3) we fix i ^ j. Denote Si = (Sx,Si,Si3). It can be assumed that without loss of generality S;xS=Ss. It is convenient to introduce:
ak ,i =
a + \k +1\2 ^a + \k -1\2
ßiki =
sfa
(12)
Using (3), (6) we see
ad gt ej = JJ^(a + \k\2)(a + |l|2) P-(coskxcosix)St
■ ÄkÄ,V (a + \k\ 2)(a + |l|2)1 -f
cos(k +1) cos(k -1)
a
(a + И2)
ad;.e/ =ÄkÄi,
Ä
Äk+iak ,l
eL +
+ \(k +1 )\ Ä
a
+ \(k -1 )\
J ß k-l
Jk-l ßk ,l
a +
cos kx cos lxS =
a + l'
f
= —Ja + \k\2 2 y
J J a + k +1
-k+l
J.
a+
k -1
k-l
Äk+l\la + \l\
Äk - l\la +
f J^kl s , Jßkl s
+ e—
\
\ Äk+l
J
■k-l J
(13)
Further, we have
ad 4 fi = JJA(a + \k\2)(a + lll2) P- (cos kxsin lx) Ss
J
(a + \k\2)
Г Ä Ä Л TJ—f ш +s(k -1)/=
\ Jk+iak ,l Jk-l ßk ,l J
if k Ф l,
ad f = - fyj(a + \k\f2k ;
J2k^k ,k
s
2
Civil Aviation High Technologies
Vol. 20, No. 02, 2017
ad f =-k-
+ik2
= (a + \k\2)
■ + |i|2
-ak ,i V -k+i
cos kxsin ix 8„ =
-ßki
fk+ - -lßL e(k -1)f
-k-i
k-l
, if k ф l,
ad If =-^(a + \k\2) flk.
-
'2 k
(14)
We have also
ad^ f/ = —V(a + |k| )(a +|i| ) Pa 1 (sin kxsin ix) 8s
-
V(a + И2)
-
-
л
- a - ß k-i
/lk+iak,i —k-i ßk,i У
ad ff =-k-
a +
k
sin kxsin ix 8 =
= -S/ (a + H2)
a + i'
f -
a
V -k+i
ek+i +
- ß*Jc,_
--i ek-i У
(15)
And further,
s
s
л
2
2
ad fiel =-ad ejfk =
^ (a + k2)
f
_-_fs__-_s(k -1) fJL
л J k + i л V У t/ j
-k+iak ,i -k-i ßk ,i
л
if k ф i,
-
ad f-ek (a + |k| ^
1
-2kak ,k
fS •
J 2k •
ad *, te! =-k-
a +
|k|2
^ (a + kl 2>
a + |i|
f
sin kxcos ix 8 =
-ak,i rs -ßk,i ,ч r
— fk+i +— <k - i f-i
Л
V k+i
k-l
if k Ф i,
ad ff =^V(a + |k|2) Ol f2k.
2 -2k
(16)
2
Remark 1. From the method of indexing elements of the basis (9) specified above, codes k +1 in the formulas (13)—(16) for different vectors l provide different elements of [Z3], and therefore, they
correspond to different elements of the basis (9). To index k -1 one element se [Z3] may in certain cases correspond to exactly two different vectors l, and, consequently, to two different elements of the
Vol. 20, No. 02, 2017
Oivil Aviation High Technologies
basis (9). This is possible if k -11= k -11 = se [Z ]; in this case denote l1 = k -s . Then for
i2 = k + sФlj we have k -12 =-s=s . Denote
Mk, i = ak, i , vk, i = ßk, i
a
k ,i
ßk
k ,i
(17)
Using the formulas (13)-(16) we see that the operator (8) has the form:
S4e/ = J^ (a + k2)
J s J s
2 Mk,l ek+l Vk,iejZi
V k +l
SJi =
ekl 4
=
ek
^fM)
т^+Ю Jo f
J
л
7 Mk,lfk+l 7 vk,lf~i V Äk+1 Äk -1
if k Ф l;
S J = jS/ (a + k2)
Ji s Jl s Mk,l ek+l Vk,ie= V Äk+l Äk-l J
Skel =
W =
T^(a + 1 k2 )т Mk kfk
л
Äl rS Äl J,s
Mk, lJk+l +~n vk, lJ=i
\Äk+1 Äk -1 J
if k Ф l;
(18)
(19)
Then we transform the expressions for /ukl, vkl.
Mk, i =a
к l
2(k ,l) + | k|2
Vkj =ßK, 1 -
2 к, 1 J (a + \k +1\ 2)(a +1/|2) ' - 2(k, l) + |k|2 ßKi J (a + \k -l\ 2)(a + ZI2)
(20)
We obtain an asymptotic form of the formulas (20) at |l| ^
да.
a + \k
+1\2 )(a V''2
1 +
2 (k, l)++a
2 I, i2 I, i2
v л
(M
2 i,|4
( . \
+ О
1 +
a
JV I I J
ijca+lk
(a V12
|l| 3
1 - 2M.+Ikf+
v л
1 ( k, l)
T + + o
a
1 +
jv 11 J
vi 1 J
V
3
V П J
1
1
1
1
2
1
1
Civil Aviation High Technologies
Hence
Vol. 20, No. 02, 2017
1 (k, l ) 1 _ 2(k, l ) \k\2 2(k, l )2 1
^ = (2(k, l) + | k| 2)(^ - + оф)=^^kF +
+ o(^tX
/ ^/7 74 I7|2w 1 (k,l) , 1 ^ 2(k,l) Ikl 2(k,l)2 . 1 . vk l = (—2(k, l) + k ) (— ++ o(—— ))=—Ц-^ + Ц---Цт^ + o(—
k( ( ' ) 11 )(|l|2 |l|4 (|l|3 \l\2 |l|2 |l|4 (|l|2
2 |l|4 22
4 ~ ' o(l,|2
(21)
Note that for |k| <|l| we have , and also ju0l = voz = 0. Therefore, we consider
the case k ^ 0 for investigation of the asymptotic behavior (20). Denote
Л =
\vol(T3)
^ = 4, ak,l =
2 (k, l) u _\k\2 2 ( k, l )2
,2 , ,l = ,, ,2 "
We obtain the following asymptotic formulas:
V/ -4V (a + \k\2)
^ л
(ak ,l + ,l + K^t)) eLl + (— ak ,l + bk ,l + o(^))e
k—l
f/ = 4V (a + Щ2)
/ Л
(ak,l + bkl + K^r)) fks+i + (ak,l — bkl + obT))f=
и -
и -
k—l
S ej = 4V (a + lkl2)
fk l 4
Г 1 1 ^
(ak, l + bK l + ./k+z + (—ak, l + l + ^j1 ))f=
rl rl
k—l
S f = 4V (a + k2)
1
1
(—ak,l — bkl + o(-2)) ek+l + (—ak,l + bkl + o(-2))eS
и '
и '
k—l
(22)
Proposision 1. The operators S ; ,Sare compact in the sense of the L -metric (4).
ek Jk
This follows from the formulas (21)-(22), Remark 1 and [6, p. 234].
Proposition 2. The norms of operators S ;, Sin the sense of the L2 -metric (4) are evaluated
ek fk
as follows
Si
S fk
<
4(42+2) k| )
(a + k )
a
(23)
To prove this, we estimate the components of (20).
2
S
e
k
k
2(k, l )
yj(a + |k +1| 2)(a + |l|2)
2
<
(k,l )
+
|k +1\ )
4~a '
Vol. 20, No. 02, 2017
Civil Aviation High Technologies
II I |2 I 12
I 12 I 12 I 12 I 12 I i2 I i2 I и I i i k t \k\ \k\
Note that \k +1\ +1/| = |k\ + 2|l\ + 2(k, l) > |k\ + 2|l\ - 2\k\l\ = 2(|l| - ^)2 + ■Щ>.
Consequently, we have
|k|2
<
w2
^a + |k +1\2)(a + |l|2)" ПЦ
< k'J2. Then we obtain the estima-
tions \Vk/I, kkJ< (2 + .
In conclusion, we use Remark 1 and the formulas (18), (19).
Corollary 1. The coefficients gki (t), hki (t) of the expansion (10) in the orthonormal basis (9) of a solution m(t) of Landau-Lifshitz equation satisfy Lipschitz condition with respect to t with the Lipschitz constant:
l(4l + 2) k\)
(a + k ).. Il2
(-1 m(0) .
a
(24)
Theorem 1. For any element m e V(T3) the operator Sm has a finite norm in the sense of
2
L -metric (4).
Proof. Represent m in the form (10).
m
=Z g о,,- e0 + Z( gk, ek + к л).
(25)
ke[Z3], i
Since the vector field m has class Cœ, then from the expansion properties of the Fourier series [7] and the fact that elements of (9) differ from element of standard Fourier-basis multiplier of the order 1
we have that for any natural n there exist such Cn > 0, Nn > 0 that the following estimations hold:
|gk,iI ,|hk,,I<,ke[Z3],\k\>Nn. k
(26)
It follows that
S ,
gk,, ek
S
hkifk
C №
A(V2 + 2) k| )
(a + k )
a
Using (26) for n > 6 we see that the series
sm =Zs i + Z (S , + S, , )
m ^ go,i e0 V gk i e'k hk f ;
i ke[Z3], i
2
Civil Aviation High Technologies
Vol. 20, No. 02, 2017
converges in the norm of the linear operator.
CONCLUSION
For the Landau-Lifshitz equation in the three-dimensional torus it is established that the operator Sm (8) has a finite norm, though it is not true the operators of the adjoint action adm and coadjoint
action ad*m of the final regulations. It follows that the coefficients of expansion of the solution in an orthonormal basis of eigenvectors of the Laplace operator satisfy Lipschitz conditions. Thus, for the Landau-Lifshitz equation on the three-dimensional torus situation is similar to the equations of an ideal fluid and Korteweg de Vries. On the other hand, if for the equations of fluid dynamics and Korteweg de Vries, this fact has been established in a general way, for the Landau-Lifshitz equation in the three-dimensional torus it is obtained specifically through the calculation of structural constants and the matrix of the coadjoint action for the current algebra with non-standard Lie bracket.
REFERENCES
1. Arnold V.I., Khesin B.A. Topological methods in gydrodynamics. Springer, New-York, 1998, 392 p.
2. Khesin B.A., Wendt R. The geometry of infinite-dimensional group. Springer. New-York,
2009, 304 p.
3. Aleksovskii V.A., Lukatskii A.M. Nonlinear dynamics of the magnetization of ferromag-nets and motion of a generalized solid with flow group. Theoret. and Math. Phys., 1990, vol. 85, no. 1, pp. 1090-1096.
4. Lukatsky A.M. Structural and geometric properties infinite Lie groups in the application of the equations of mathematical physics. Yaroslavl, Yaroslavl State University named P.G. Demidov,
2010, 175 p. (in Russian)
5. Lukatsky A.M. Investigation of the geodesic flow on an infinite-dimensional Lie group by means of the coadjoint action operator. Proceedings of the Steklov Institute of Mathematics, December 2009, vol. 267, issue 1, pp. 195-204. (in Russian)
6. Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. M., Nauka, 1972, 496 p. (in Russian)
7. Zhuk V.V., Natanson G.I. Trigonometric Fourier series and elements approximation theory. Leningrad. Publishing House of Leningrad University Press, 1983, 188 p. (in Russian)
8. Temam R. Navier-Stokes equations. Theory and numerical analysis. North Holland Publ. Comp., 1979.
9. Arnold V.I. Mathematical methods in classical mechanics. M., Editorial URSS, 2000, 408 p. (in Russian)
10. Zulanke R., Witten P. Differential geometry and fiber bundles. M., Mir, 1975. (in Russian)
INFORMATION ABOUT THE AUTHOR
Alexander M. Lukatsky, a leading researcher at the Energy Research Institute of Russian Academy of Sciences, Doctor of phys.-math. sciences, [email protected].
О СТРУКТУРЕ ДЕЙСТВИЯ КОПРИСОЕДИНЕННОГО ОПЕРАТОРА НА АЛГЕБРЕ ТОКОВ ТРЕХМЕРНОГО ТОРА
А.М. Лукацкий1
1 Институт энергетических исследований РАН, г. Москва, Россия
Vol. 20, No. 02, 2017
Civil Aviation High Technologies
Для алгебры Ли потоков на трехмерном торе с нестандартной скобкой Ли установлены некоторые свойства, в случае когда сумма присоединённого и коприсоединенного операторов на бесконечномерной алгебре Ли со
скалярным произведением, имеет конечную норму. Точнее, для уравнения Ландау - Лифшица на трехмерном торе
*
установлено, что оператор Sm = (adm + adm) / 2 имеет конечную норму, хотя это не так для присоединённого действия ad и коприсоединённого действия adт. Из этого выводится, что коэффициенты разложения решения по
ортонормированному базису собственных векторов оператора Лапласа удовлетворяют условию Липшица. Таким образом, для уравнения Ландау - Лифшица на трехмерном торе ситуация схожа с таковой для идеальной жидкости и уравнения Кортвега - де Фриза. С другой стороны, если для уравнений гидродинамики и уравнения Кортвега - де Фриза такой факт был установлен в общем виде, то для уравнения Ландау - Лифшица на трехмерном торе это выведено специальным способом, через вычисление структурных констант и матрицы коприсоеди-нённого действия на алгебре токов с нестандартной скобкой Ли.
Ключевые слова: алгебра токов, скобка Ли, действие присоединённого оператора, оператор коприсоеди-нённого действия, трёхмерный тор, уравнение Ландау - Лифшица, компактный оператор, условие Липшица.
СПИСОК ЛИТЕРАТУРЫ
1. Arnold V.I., Khesin B.A. Topological methods in gydrodynamics. Springer, New-York, 1998, 392 p.
2. Khesin B.A., Wendt R. The geometry of infinite-dimensional group. Springer. New-York, 2009, 304 p.
3. Aleksovskii V.A., Lukatskii A.M. Nonlinear dynamics of the magnetization of ferromag-nets and motion of a generalized solid with flow group. Theoret. and Math. Phys., 1990, vol. 85, no. 1, pp. 1090-1096.
4. Лукацкий А.М. Структурные и геометрические свойства бесконечномерных групп Ли в приложении к уравнениям математической физики. Ярославль: Ярославский Государственный Университет, 2010. 175 с.
5. Лукацкий А.М. Исследование геодезических потоков на бесконечномерных группах Ли при помощи действия коприсоединенного оператора // Труды Математического института им. Стеклова. 2009. Т. 267, вып. 1. С. 195-204.
6. Колмогоров А.Н., Фомин С.В. Элементы теории функций и функционального анализа. M.: Наука, 1972. 496 с.
7. Жук В.В., Натанзон Г.И. Тригонометрические ряды Фурье и элементы теории аппроксимаций. Л.: Издательство Ленинградского Университета, 1983. 188 с.
8. Temam R. Navier-Stokes equations. Theory and numerical analysis. North Holland Publ. Comp., 1979.
9. Арнольд В.И. Математические методы в классической механике. М.: Эдиториал УРСС, 2000. 408 с.
10. Зуланке Р., Виттен П. Дифференциальная геометрия и расслоения. М.: Мир, 1975.
СВЕДЕНИЯ ОБ АВТОРЕ
Лукацкий Александр Михайлович, ведущий научный сотрудник Института энергетических исследований РАН, доктор физ.-мат. наук, [email protected].
