ON INDIRECT REPRESENTABILITY OF FOURTH ORDER ORDINARY DIFFERENTIAL EQUATION IN FORM OF HAMILTON-OSTROGRADSKY EQUATIONS Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 15. № 3 (2023). С. 121-131.

ON INDIRECT REPRESENTABILITY OF FOURTH ORDER ORDINARY DIFFERENTIAL EQUATION IN FORM OF HAMILTON-OSTROGRADSKY EQUATIONS

S.A. BUDOCHKINA, Т.Н. LUU, V.A. SHOKAREV

Abstract. In the paper we solve the problem on the represent ability of a fourth order ordinary differential equation in the form of Hamilton-Ostrogradskv equations. Local bilinear forms play an essential role in the investigation of the potentiality property of the considered equation. It is well known that the problem of representing differential equations in the form of Hamilton-Ostrogradskv equations is closely related to the existence of a solution to the inverse problem of the calculus of variations, that is, for a given equation one needs to construct a functional-variational principle. To solve this problem, we first obtain necessary and sufficient conditions for the given equation to admit an indirect variational formulation relative to a local bilinear form and then construct the corresponding Hamilton-Ostrogradskv action. Note that the found conditions are analogous to the Helmholtz potentiality conditions for the given ordinary differential equation. We also define the structure of the considered equation with the potential operator and use the Ostrogradskv scheme to reduce the given equation to the form of Hamilton-Ostrogradskv equations.

It should be noted that applications and extensions of the work are the possibility to establish connections between invariance of the functional and first integrals of the given equation and to extend the proposed scheme to partial differential equations and systems of such equations.

Keywords: Local bilinear form, potential operator, Hamilton-Ostrogradskv action, Hamilton-Ostrogradskv equations.

Mathematics Subject Classification: 49N45, 70H05

1. Introduction

Classical Hamiltonian formalism is employed for solving various problems of mathematics, mechanics and physics. It plays an important role in determining some approaches to the investigation of motion of different complex systems, which is motivated by very effective methods for integration and qualitative investigation of equations of motion [11], [12], [13], [15]. For this reason, the solution to the problems associated with the representation of different types of equations and their systems in the form of Hamilton equations and their generalizations is of great interest, see, for instance, [2], [5], [20].

S.A. Budochkina, Т.Н. Luu, V.A. Shokarev, On indirect representability of fourth order

ordinary differential equation in form of hamilton-ostrogradsky equations.

© Budochkina S.A., Luu Т.Н., Shokarev V.A. 2023.

This publication has been partially supported by the RUDN University named after Patrice Lumumba, project no. 002092-0-000 and by the Ministry of Science and Higher Education of the Russian Federation (megagrant agreement No. 075-15-2022-1115).

Submitted April 26, 2022.

These issues are closely related to the inverse problem of the calculus of variations (IPCV) in the following statement: for a given equation, one needs to construct a functional such that its set of stationary points coincides with the set of solutions to this equation. There is a large number of works devoted to inverse problems of the calculus of variations: for ordinary differential equations and partial differential equations [3], [4], [7], [9], [19], [20], [26], [27], operator equations [6], [21], [22], differential-difference equations [8], [17], [18], stochastic differential equations [23], [24], [25], fractional differential equations [1], [10], [14], [28], In these works, nonlocal bilinear forms were mainly used to solve the IPCV,

The representabilitv of a given equation with a non-potential operator in the form of Hamilton equations can be a difficult problem. One of the ways for solving it is to use local bilinear forms. The main aim of the paper is to establish connection between variationalitv of a fourth order ordinary differential equation and Hamilton-Ostrogradskv equations, i.e., to obtain the conditions of the indirect representabilitv of the given equation in the form of the Lagrange-Ostrogradskv equation, to construct the corresponding funetional-variational principle, to define the variational structure of the considered equation and to apply the Ostrogradskv scheme [16] for the representation of this equation in the form of Hamilton-Ostrogradskv equations. The method of constructing functional developed in [3] is extended to a fourth order ordinary differential equation. Local bilinear forms play a significant role in our study.

In what follows we use the notation and terminology of [3], [20], Let U, V be real linear normed spaces. The following definition and theorem will be employed in what follows.

Definition 1.1. ([20]) An operator N : D(N) C U ^ V is called potential on a set D(N) relative to a local bilinear form $(u; ■, ■) : V x V ^ R if there exists a Gateaux differentiable functional Fn : D(F^) = D(N) ^ R such that

6Fn[u, h] = $(u; N(u),h) Vu e D(N), Vh e D(N'U).

Theorem 1.1. ([20]) Consider a Gateaux differentiable operator N : D(N) C U ^ V and a local bilinear form $(u; ■, ■) : V xV ^ R such that for all fixed elements u e D(N), g,h e D(N'U) the function ^>(e) = $(« + eh; N(u + eh),g) belongs to the class C:[0,1]. Then the operator N is potential on the convex set D(N) relative to $ if and only if

$ (u; N'uh, g) + $'u (h; N(u),g) = $ (u; N'ug, h) + $'u (g; N(u),h) (1.1)

for all u e D (N), h,g e D (N^). Under this condition the potential F^ is given by

i

Fn[u] = J $(uo + X(u — u0); N(u0 + X(u — u0)),u — u0) dX + FN[u0], (1.2)

o

where u0 is a fixed element of D(N).

2. On indirect variational formulation of fourth order ordinary

DIFFERENTIAL EQUATION We consider a fourth order ordinary differential equation

4

N(u) = ^ai(t,u(t))u{i)(t) + ao(t,u(t)) = 0, t e (t0,t1). (2.1)

i=1

Here u = u(t) is an unknown function, a» e Ct([t0,t1] x T), i = 1, 4, a0 e C 1([t0,t1] x T) are given functions, T C R, a4(t,u(t)) = 0.

We define the domain of the operator N as follows:

D(N) = {u e U = C4[t0,h] : u(t0) = uu u(t1) = u2, u'(¿0) = u[, u'(t1) = u'2} . (2.2)

We observe that V = C[t0,t1] and

D(N'U) = {h e U = C4[to,ti] : h(to) = 0, h(U) = 0, h'(to) = 0, ^'(i!) = 0} . We introduce a local bilinear form by

ti

$(u; v,g) = J M(t,u(t))v(t)g(t)dt, (2.3)

to

where M e C4([io,ii] x T), M(t,u(t)) = 0. We denote

Mi(t,u(t)) = M (t,u(t))ai(t,u(t)), i = 0,4.

Theorem 2.1. The operator N is potential on D(N) relative to bilinear form (2.3) if and only if Mi = Mi(t), i = 1, 4 and for all t e [t0,t!] the following conditions hold:

M3(t) = 2M4(t), (2.4)

Mi(t) = M'2(t) - M'l'(t). (2.5)

Proof. We have

44

Kh =Ys <n(t, u(t))h(t)u(i (t) + a'ou(t, u(t))h(t) + £ Oi(t, u(t))h(i (t).

i=1 i=1

In this case potentiality criterion (1.1) becomes

ti

M (t, u(t)) J] a'iu(t, u(t))h(t)u(i\t)g(t) + M (t, u(t))a'ou(t, u(t))h(t)g(t)

to \ i=1

+ M(t,u(t)) ^ ai(t)u(t))h(i)(t)g(t)\ dt

+

o

i=i )

i 4

' (i)

j ^M'u(t,u(t)) ^ ai(t,u(t))u(i)(t)h(t)g(t) + M'u(t,u(t))a0(t,u(t))h(t)g(t^ dt

j (M(t, u(t)) ^ aiu(t, u(t))g(t)u(i') (t)h(t) + M(t, u(t))a'ou(t, u(t))g(t)h(t)

to \ i=1

+ M(t, u(t)) ^ ai(t, u(t))g(i) (t)h(t)\ dt

i=1 J

J ^M'u(t, u(t)) ^ ai(t, u(t))u(i\t)g(t)h(t) + M'u(t, u(t))a0(t, u(t))g(t)h(t^j dt

(2.6)

ti 4 + / I W dud)) ^ n .(f

to

for all u e D(N), h,g e D(N'u). Hence

tl 4 tl

i)i

M(t, u(t))J2 ai(t, u(t))h(i) (t)g(t)dt = M(t, u(t))^ ai(t, u(t))g(i) (t)h(t)dt

, i=1 , i=1

to to

for all u E D(N), h,g E D(N^), or

J Mi(t, u(t))h(i\t) g(t)dt = J Mi(t, u(t))g(i\t)h(t)dt

(2.7)

for all u E D(N), h,g E D(N'U). Integrating by parts and taking into consideration that

^ Mi(t, u(t))h(i) (t)g(t)dt = h(t)( - M'u(t, u(t))g(t) - M[u(t, u(t))v!(t)g(t)

- M1(t,u(t))g'(t) + MZtt(t,u(t))g(t)

+ 2M'2tu(t, u(t))u' (t)g(t) + M'2uu(t,u(t))(u' (t))2g(t) + M'u(t,u(t))u" (t)g(t) + 2M't(t,u(t))g' (t) + 2M2u(t,u(t))u! (t)g'(t) + M2(t,u(t))g"(t)

- M'3itt(t,u(t))g(t) - 3M"itu(t, u(t))u' (t)g(t)

- 3M'tt(t,u(t))g'(t) - 3MZau(t,u(t))(u'(t))2g(t)

- 3M"tu(t, u(t))u"(t)g(t) - 6M'tu(t, u(t))u'(t)g'(t)

- MZuu(t,u(t))(u!(t))3g(t) - 3M'uu(t,u(t))ul(t)u"(t)g(t)

- 3M''uu(t,u(t))(u'(t))2g'(t) - M'u(t, u(t))u"'(t)g(t)

- 3M'u(t, u(t))u"(t)g'(t) - 3M't(t, u(t))g"(t)

- 3M'u(t,u(t))u'(t)g"(t) - M3(t,u(t))gm(t) + Mi^(t,u(t))g(t) + 4M^t(t,u(t))g' (t)

+ 4M^ttu(t,u(t))u' (t)g(t) + 6M£lu(t,u(t))(u' (t))2g(t) + 6MH;tu(t,u(t))u"(t)g(t) + 12M'lltu(t, u(t))u'(t)g'(t) + 6MHtt(t,u(t))g''(t) + 4Mifuu(t,u(t))(u' (t))3g(t) + 12M'H-uu(t, u(t))u' (t)u" (t)g(t) + 12M%uu(t,u(t))(u' (t))2g' (t) + 4M'ltu(t, u(t))u"' (t)g(t) + 12MUt,u(t))u"(t)g'(t)

+ 12M'tu(t, u(t))u'(t)g''(t) + M^uuuttMtW(t))49(t) + 6MZuu (t,u(t))(u' (t))2u" (t)g(t) + 4MZuu(t,u(t))(u! (t))3g' (t) + 3M'luu(t, u(t))(u"(t))2g(t) + 4Mlu(t,u(t))u>(t)u"'(t)g(t) + 12 Mlu(t ,u(t))u' (t)u" (t)g'(t) + 6M%uu(t ,u(t))(u' (t))2g''(t) + M'u(t,u(t))uw(t)g(t) + 4M[,a(t, u(t))u"'(t)g'(t) + 6MKt, u(t))u"(t)g"(t) + 4M'At(t, u(t))g"'(t)

+ 4 M[u(t,u(t))u'(t)g"'(t) + M4(t,u(t))g(4)(t)^ dt.

- M'u(t,u(t)) - M[u(t,u(t))u'(t) + M2tt(t,u(t)) + 2M'2tu(t,u(t))u'(t)

+ M'uu(t,u(t))(u'(t))2 + M'u(t,u(t))u"(t) - M^(t,u(t)) - 3M^u(tMt))u'(t)

- 3M£uu(t,u(t))(u'(t))2 - 3MHtu(t,u(t))u"(t) - MZuu(t,u(t))(u'(t))3

h,g E D(Nu), we get

Therefore, it follows from (2.7) that

- 3M3uu(t,u(t))u'(t)u"(t) - M'3u(t,u(t))u"'(t) + M4utt(t,u(t)) + 4M^)tu(t,u(t))u'(t) + 6 M4£uu(t ,u(t))(u' (t))2 + 6M44ltu(t ,u(t))u" (t) + 4M(42iuu(t ,u(t))(u' (t))3 + 12 MZuu(t ,u(t))u' (t)u" (t) + 4M4tu(t ,u(t))u"'(t) + M4£uuu(t ,u(t))(u' (t))4 + 6 MZuu (t ,u(t))(u' (t))2u" (t) + 4Mlu(t ,u(t))u' (t)u"'(t) + 3 Mlu(t ,u(t))(u" (t))2 + M4u(t ,u(t))u(4\t) = 0,

and

- 2M1(t,u(t)) + 2M2t(t,u(t)) + 2M2u(t,u(t))u'(t) - 3M'3tt(t,u(t)) - 6M'3tu(t,u(t))u'(t)

- 3M'iuu(t,u(t))(u'(t))2 - 3M3u(t,u(t))u"(t) + 4Mdtt(t,u(t)) + 12Mgtu(t,u(t))u'(t) + 12 M4iuu(t ,u(t))(u' (t))2 + 12M4tu(t ,u(t))u"(t) + 4MZuu(t ,u(t))(u' (t))3

+ 12 M4uu(t ,u(t))u' (t)u" (t) + 4M4u(t ,u(t))u"'(t) = 0,

and

- M3t(t,u(t)) - M3u(t,u(t))u'(t) + 2M4tt(t,u(t)) + 4M4tu(t,u(t))u'(t) + 2 Mlu(t ,u(t))(u' (t))2 + 2M4u(t ,u(t))u" (t) = 0,

- M3(t, u(t)) + 2M4t(t, u(t)) + 2M4u(t, u(t))u'(t) = 0. This yields that Mi = Mi(t), i= 1,4, and

- M3(t) + 2M4(t) = 0, (2.8)

- 2Ml(t) + 2M2(t) - 3M'i(t) + 4M4'(t) = 0, (2.9)

- M[(t) + M'2, (t) - M'3'(t) + M(4)(t) = 0. (2.10)

We observe that conditions (2.8)-(2.10) are reduced to (2.4) and (2.5) and this completes the proof. □

3. Construction of Hamilton-Ostrogradsky action

Theorem 3.1. If the operator N is potential on D(N) relative to bilinear form (2.3), then the corresponding Hamilton-Ostrogradsky action is given by ti

Fn [U] = i

1 (M'I(t) - M2(t)) (u'(t))2 + 1M4(t)(u"(t))2 + BM(tMt))

dt, (3.1)

to

where

BM(t,u(t)) = J M0(t,u(t,\))(u(t)-ua(t))d\ + Bm (t, uo(t)), (3.2)

0

u(t, X) = u0(t) + X(u(t) — u0(t)), u0 = u0(t) is a fixed element of D(N), BM E C2([t0, t\] x T).

Proof. Using formula (1.2) and potentiality conditions (2.4), (2.5), we get: ti 1

Fn[u] — Fn[uo] = J J (Mo(t,u(t, X))(u(t) — uo(t)) + Mi(t)Ut(t,X)(u(t) — uo(t))

o 0

+ M2(t)u'lt(t, X)(u(t) — uo(t)) + M3(t)ÛC(t, X)(u(t) — uo(t))

1

+ MXt)u¿t(t, л)(и(г) - Uo(t))) с1ЛсИ ti i

j J (Mo(t,u(t, Л))(и(Ь) - Uo(t)) + Mi^v^t, Л)(и(Ь) -Uo(t))

Q o

- М2(г)и№, Л)(и(t) - u0(t)) - M2(t)u'Xt, Л)(иЦ) - u0(t))'

- М'з^иЪ(t, лx^t) - Uo(t)) - Мз(г)и^(t, лx^t) - Uo(t))'

- м[(^и'Ш, л)(и(г) - u0(t)) - м4, л)(и(г) - u0(t))') (1Л(И

ti i

= J j (Mo(t ,u(t ,Л))(и(1) - Uo(t)) + MXtWXt, Л)(и(Ь) -Uo(t))

Q o

- , л)(и(г) - u0(t)) - M2(t)mt, л)(и(г) - u0(t))'

+ M', ЛХи(1) - Uo(t)) + M'^u^t, ЛХи(1) - Uo(t))'

- M*(t)^(t, лХи(г) - Uo(t))' + Ml(t)u'lt(t, лХи(г) - uo(t))

+ 2МЦ^и'ы(t, Л)(и(г) - Uo(t))' + М^)и'и(t, Л)(и(г) - uo(t))'') (1Л(И

ti i

= JJ (Mo(t,u(t, Л))(и(Ь) - Uo(t)) + MX^t, Л)(и(Ь) -Uo(t))

Q o

- , л)(и(г) - uo(t)) - М2(г)и'(, л)(и(г) - uo(t))' + М'ЦЩ^, Л)(и(1) - Uo(t)) + M's(t)u'Xt, Л)(и(1) - Uott))'

- M,(tyü'lt(t, ЛХиЦ) - Uo(t))' - МЦ'^ЦЦ, ЛХиЦ) - Uo(t))

- Ml(t^Xt, ЛХи(Ь) - Uo(t))' + 2MXt)u"t(t, ЛХи(Ь) - Uo(t))'

+ MXt)u'it(t, л)(и(г) - u0(t))'') ¿Леи

ti i

= J j (iïxt,л)(и(г) -МЩмхг) -м2(г) + м'(t) -M^t))

Q o

+ Mo(t,u(t,Л)Хи(г) -Uo(t)) + %(t, лхи(г) - uoW( - M2(t) + M'(t) - Ml(t)) + (t, лхи(г) - Uo(t))'( - M"(t) + 2M4(t)) + MXt)ù"t(t, л)(и(г) - uo(t))'') ¿Леи

ti i

= jj (u'Xt, Л)(и(Ь) - uo(t)y (Ml(t) -M2(t))

tQ 0

+ MXt)ù"t(t,л)(и(г) -Uo^y + Mo(t,u(t,л)Хи(г) -uo(t))) йллъ.

We observe that

i i

J M4 (t)^ (t ,лхи(г)-щ^хчл = J M4(t)v!¿t (t ,Л)Г4Х(1 ,л)<!Л

oo

i

= 1f Ж(м>Л))2) ^

= l-Mi(t)(^'(t))2 - ^2Mi(t)(u'l(t))2, 1 1

J (M''(t) - M2(t)) u't(t, X)(u(t) - Uo(t))'dX = j (M'i(t) - M2(t)) u't(t, X)u'lx(t, X)dX 0 0

i

((M'i(t) - M2(t)) (u't(t,X))2) dX

Therefore,

ti

<-1 / /

to V

1 f JL (( M'.'(i) — (+)) (rf. (+ w2>

2 J dX

0

12

- (M'l(t) - M2(t)) (u'(t))2

- 1(M'i(t) - M2(t)) (u'o(t))2 .

1 ' ~ ' '" ' ' ' s s ' " - ^ 2 1 I , «-// , r I , N N /!/ ,\\2

Fn[u] - Fn[uo] = I ( 2 (M'l(t) - M2(t)) (u'(t))2 - - (M'l(t) - M2(t)) (u'o(t))

+ \m4(t) (u"(t))2 - \ma(t) (u'0(t))2

i

^y Mo(t,u(t,X))(u(t) -uo(t))dX ) dt.

Taking into consideration (3.2), we obtain Hamilton-Ostrogradsky action (3.1). The proof is complete. □

vj dt.

4. On variational structure of equation (2.1)

Theorem 4.1. The operator N is potential on D(N) relative to bilinear form (2.3) if and only if equation (2.1) reads as

2 M' (t)

N(u) = a4(t,u(t))u(4)(t)+ 4(/.. u"'(t) + a2(t,u(t))u"(t)

M (t ,u(t))

+M2(t) - M'"(t) , + (BM)u(t,u(t)) (.)

+ M (t ,u(t)) ()+ M (t ,u(t)) 0.

Proof. Let us find a variation of Hamilton-Ostrogradsky action (3.1). We have

ti

5Fn[u, h]= j (( M'l(t) - M2(t)) u'(t)h'(t) + MA(t)u"(t)h''(t) + (BMYu(t,u(t))h(t)) dt

o ti

= j ( - M'i'(t)u'(t)h(t) - M'i(t)u"(t)h(t) + M2(t)u'(t)h(t) + M2(t)u"(t)h(t)

to

- M[(t)u"(t)h'(t) - MA(t)u"'(t)h'(t) + (Bmyu(t,u(t))h(t)) dt

ti

= j ( - M'i'(t)u'(t)h(t) - M'i(t)u"(t)h(t) + M2(t)u'(t)h(t) + M2(t)u"(t)h(t)

o

+ MH(t)u"(t)h(t) + 2M'a(t)u"'(t)h(t) + M4(t)u[A)(t)h(t) + (BmYu(t,u(t))h(t)) dt

/2 M' (t)

M(t,u(t))(a4(t,u(t))u(4)(t) + (/.. u"'(t) + c(t,u(t))u"(t)

M (t ,u(t))

for all u E D(N), h E D(Nu). Hence, equation (2.1) is represented in form (4.1). On the other hand, equation (4.1) is derived from the stationarity condition of Hamilton-Ostrogradsky action (3.1). It means that conditions (2.4), (2.5) must be satisfied. □

5. On connection between variationality of equation (2.1) and Hamilton-Ostrogradsky equations

Theorem 5.1. If operator N (2.1) is potential on D(N) (2.2) relative to bilinear form (2.3) then equation (2.1) is represented in the form of Hamilton-Ostrogradsky equations

u' (t) = u' (t), u"(t) = ,

W W' W M4 (t) (5.1)

P'i(t) = (Bm)'u(t,u(t)), p2(t) = (M'i(t) - M2(t))u'(t) - pi(t),

where

Pi(t) = (M'l(t) - M2(t))u'(t) - M'4(t)u"(t) - MA(t)u"'(t), p2(t) = MA(t)u"(t). Proof. From (3.1) we obtain the Lagrangian

L[t,u(t),u'(t),u"(t)] = 1 (M'l(t) - M2(t)) (u'(t))2 + 1m4(t)(u"(t))2 + Bm(t,u(t)).

Then

L

P2(t) = ^l~) = M4 (t)u" (t) (5.2)

and

P2(t)

u" (t)

M4 (t)'

Hence, the Hamiltonian is

H[t,u(t),u'(t),pi(t),p2(t)] =pi(t)u'(t) + (p2(t)u"(t) - L[t,u(t),u'(t),u"(i)]^

u''{t)

= P2(t) ' M4(t)

--Pi (t)u'(t) + ( P2(t)v!' (t) - 1( M'Kt) - M2(t))(u' (t))2

- (t)(u" (t))2 -Bm (t ,u(t))

1 P2(t) 1

)

u"(t)- P2(t) u (l)— M4(t)

We have

that is,

=Pi (t)u'(t) + - - 2(M'l (t) - M2(t))(u> (t))2 - Bm (t ,u(t))

dH P2(t)

dp 2 (t) M4 (t)' dH

dp 2 (t)

= u" (t).

Then

dH dH dH

u' (t), = -(BM )'u(t ,u(t)), = P1(t) - (Ml (t) - M2(t))u' (t).

dp 1 (t) du(t) K MJu^ w" du'(t)

We note that for the Hamilton-Ostrogradsky action

ti

Fn [u] = j L[t ,u(t),u' (t),u" (t)]dt

o

the corresponding Lagrange-Ostrogradsky equation is

dL d dL d2 dL N(u) = - ^^tttv +

or

For

du(t) dt du'(t) dt2du"(t) d ( dL d dL \ dL

d f dL d dL \ = dL ( )

dt \du' (t) - Jtdu" (t)J = du(t). (.)

d L d L

P1(t) = - ~T, -TT-^ = (Ml(t) - M2(t))u'(t) - M4(t)u"(t) - M4(t)u"'(t) (5.4)

du'(t) dt du" (t) from (5.3) we get

d L

P[(t) =

i.e.

Thus,

du(t)'

p\(t) = (BM )'u(t ,u(t)).

dH

P\(t) = -

du(t)'

For p2(t) (5.2) from (5.4) we have

that is

d L

p2(t) = dMy) - P1(f) = (M4(f) - M2(t))u'(t) - P1M,

d H

p2(t) = -

du' (t)' Hence,

d H d H H H

u (t) = ^T* , u '(t) = TwTn , №) = TIN , P2(t) = -

dp1(ty w dp2(ty du(ty du' (t)'

It means that equation (2.1) with the potential operator N is represented in the form of Hamilton-Ostrogradsky equations (5.1). □

0

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Svetlana Aleksandrovna Budochkina, RUDN University, Miklukho-Maklaya str. 6, 117198, Moscow, Russian Federation E-mail: [email protected]

Thi Huyen Luu, RUDN University, Miklukho-Maklaya str. 6, 117198, Moscow, Russian Federation E-mail: [email protected]

Vladimir Andreevich Shokarev, RUDN University, Miklukho-Maklaya str. 6, 117198, Moscow, Russian Federation E-mail: [email protected]