c Allerton Press, Inc., 2011. ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2011, Vol. 55, No. 9, pp. 24–36.  c V.P. Derevenskii, 2011, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2011, No. 9, pp. 30–43. Original Russian Text 

Trigonometric Solutions of Nonlinear First-Order Ordinary Differential Equations over a Banach Algebra V. P. Derevenskii1* 1

Kazan State Architecture and Building University, ul. Zelyonaya 1, Kazan, 420043 Russia Received July 12, 2010

Abstract—In this paper we continue the study initiated in the recent paper (see Russian Mathematics (Iz. VUZ) 50 (8), 5–17 (2006)). We determine differentiation formulas for basic trigonometric functions and describe classes of nonlinear first-order ordinary differential equations (ODE.1) over a finite-dimensional Banach algebra whose solutions are the mentioned functions. DOI: 10.3103/S1066369X11090040 Keywords and phrases: differential equations, matrices, Lie algebras.

1. The problem. Many real-world processes are described by nonlinear matrix and operator dynamic equations. The absence of explicit differentiation formulas for most elementary functions over a noncommutative set significantly complicates the study of the mentioned processes. This leads to the necessity to linearize such equations, which, in turn, decreases the reliability of the research results. Consequently, the modern theory needs formulas for the differentiation of various functions over nonabelian varieties and definitions of ODE, whose solutions are the mentioned functions. Since many processes are periodic, it is quite natural to consider trigonometric functions over an associative algebra which generalizes not only scalar, but also matrix-operator sets. Let us clarify this idea as follows. Denote by B a finite-dimensional Banach algebra with a unit element E, i.e., a complete metric space over a complex numeric field C representing an associative algebra, for which the product is continuous in each of multipliers ([2], P. 346). Let the symbol A stand for the restriction of the algebra B to the real field R and let LN stand for the N -dimensional Lie algebra over A with basic elements Eα and γ γ (Cαβ ∈ R, α, β, γ = 1, N , N < ∞). Assume that a continuous variable t structural constants Cαβ from R corresponds to a continuous element Q = Q(t) of the algebra A. Consider the basic trigonometric functions from A to A: ∞ ∞   (−1)n+1 2n−1 (−1)k 2k Q Q : Q0 = E, , cos Q = (1) sin Q = (2n − 1)! (2k)! n=1

k=0

−1

tan Q = sin Q(cos Q)

, ctg Q = (tan Q)−1 = cos Q(sin Q)−1 , −1

sec Q = (cos Q)

−1

, cosec Q = (sin Q)

,

(2) (3)

where formulas (2) and (3) are evaluated at points, where functions (sin Q)−1 and (cos Q)−1 are defined. It is required to find derivatives of functions (1)–(3) and to establish forms of ODE.1, whose solutions are the mentioned functions. Analogously to the scalar case, the connection between basic hyperbolic and trigonometric functions over B ([1], proposition 3) allows us to write sin Q = −i sinh(iQ) and cos Q = cosh(iQ). These equalities imply that some issues related to the stated problem can be established with the help of results obtained in [1]. However, this leads to the loss of operation principles for functions under consideration, therefore, it makes sense to solve the stated problem in accordance with the initial definitions. 2. Some properties of trigonometric functions over A immediately follow from their definitions. However, prior to describe these properties, let us mention the evident convergence (in the norm) of both series in (1) for any bounded in the norm family Q. The first property can be formulated as follows. *

E-mail: [email protected].

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25

Proposition 1. All introduced trigonometric functions commute with each other. Proof. We have [sin Q, cos Q] ≡ sin Q cos Q − cos Q sin Q    ∞ ∞ ∞ ∞ (−1)n+1 2n−1  (−1)k 2k (−1)n+1  (−1)k 2n−1 2k = Q Q [Q , , Q ] = 0. = (2n − 1)! (2k)! (2n − 1)! (2k)! n=1

n=1

k=0

{([A, B −1 ]

k=0

∧ ([A−1 , B −1 ]

= 0) = 0)}, we have [tan Q, cot Q] = [sin Q, sec Q] = Since ([A, B] = 0) ⇒ [sec Q, cosec Q] = 0. One can prove the commutativity of the rest functions analogously to the following case: [sin Q, tanh Q] = (sin Q)2 (cos Q)−1 − sin Q(cos Q)−1 sin Q = sin Q[sin Q, sec Q] = 0. Therefore, all fifteen pairs of considered functions commute with each other. Certain properties of scalar trigonometric functions allow us to obtain for B an analog of the L. Euler formula. Proposition 2. If Q ∈ A, then exp(iQ) = cos Q + i sin Q, i =

√ −1 ∈ C.

(4)

Proof. Note that the exponential function is defined over a Banach algebra as the series exp Z = ∞  1 n 2k = (−1)k Q2k and Z 2k−1 = i(−1)k+1 Q2k−1 . n! Z . Therefore, for Z = iQ, where Q ∈ A, we have Z n=0

Thus, the sum of even terms in the exponential series gives cos Q, while the sum of odd ones does i sin Q, which allows us to write formula (4). In order to establish an analog of the Pythagorean trigonometric identity, we need the following lemma. Lemma. Let Q and R be two noncommutative elements of the algebra A, then cos QR cos Q + sin QR sin Q = cos(adQ R), sin QR cos Q − cos QR sin Q = sin(adQ R),

(5) (6)

where adQ is the associated mapping operator ([4], P. 18) defined by the element Q : adQ R ≡ [Q, R]. Proof. For elements of an associative ring, whose particular case is the algebra B, the following Baker– Hausdorff formula ([5], P. 655) is valid: ∞  1 adnZ Y. exp ZY exp(−Z) = n! n=0

(7)

By putting Z = iQ and Y = R, where (Q, R) ⊂ A, from the above formula we get exp(iQ)R exp(−iQ) = exp(adiQ R) =

∞  1 adniQ R = cos(adQ R) + i sin(adQ R). n! n=0

On the other hand, in accordance with Proposition 2 we have exp(iQ)R exp(−iQ) = (cos Q + i sin Q)R(cos Q − i sin Q). Since the multiplication in B is distributive, we can write the above correlation as follows: cos QR cos Q + sin QR sin Q + i(sin QR cos Q − cos QR sin Q). However, in the algebra B, as well as in the field C, for Zk = Xk + iYk , k = 1, 2, where (Xk , Yk ) ⊂ A, it holds (Z1 = Z2 ) ⇔ (X1 = X2 , Y1 = Y2 ). Therefore, comparing real and imaginary parts of the right-hand sides of two obtained equalities, we get correlations (5) and (6). The latter assertion allows us to prove “the Pythagorean trigonometric identity” over A. RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011

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DEREVENSKII

Corollary. If Q ∈ A, then functions sin Q and cos Q defined in (1) are connected by the correlation sin2 Q + cos2 Q ≡ E.

(8)

Proof. If in the lemma we put R = E, then we get the implication (5) ⇒ (8), and formula (6) additionally proves the commutativity of sin Q and cos Q. Equality (8) implies formulas tan2 Q + E = sec2 Q and cot2 Q + E = cosec2 Q. Formulas that define sines and cosines of multiple arguments over the algebra A remain the same. Proposition 3. If Q ∈ A, then

  n , (−1) λ= cos nQ = 2 l=0   μ  n−1 m 2m+1 2m+1 n−2m−1 , (−1) Cn (sin Q) (cos Q) , μ= sin nQ = 2 λ 

l

Cn2l (sin Q)2l (cos Q)n−2l ,

(9)

(10)

m=0

where [q] is the integer part of the number q. Proof. Note that for Z ∈ B it holds (eZ )n = enZ . Then for Z = iQ, where Q ∈ A, in accordance with Proposition 2 we can write cos nQ + i sin nQ = (cos Q + i sin Q)n . Since sine and cosine of the same argument are commutative, in the right-hand side of this equality we can use the Newtonian binomial (cos Q + i sin Q)n =

n 

Cnk ik (sin Q)k (cos Q)n−k .

k=0

Equating the real and imaginary parts of the left- and right-hand sides of the obtained equality, we get formulas (9) and (10), and requirements 2λ ≤ n and 2μ + 1 ≤ n give constraints on λ and μ. According to definitions (1)–(3), the considered trigonometric functions have the same even properties as their scalar analogs. 3. Differentiation of trigonometric functions over the algebra A. Let Q = Q(t) be a continuously differentiable function of the variable t in A. Then Proposition 2 and formulas for the differentiation of operator exponents allow us to determine derivatives of basic trigonometric functions (1) and, consequently, those of the rest functions (2) and (3) which are connected with the first pair by algebraic correlations. The next theorem gives differentiation formulas for the considered functions. Theorem 1. If Q(t) is a continuously differentiable in the variable t element of A, then trigonometric functions (1)–(3) have the following derivatives: d sin Q = (cos Q)R + (sin Q)S, (11) dt d cos Q = −(sin Q)R + (cos Q)S, (12) dt d tan Q = (cos Q)−1 R(cos Q)−1 , (13) dt d cot Q = −(sin Q)−1 R(sin Q)−1 , (14) dt d sec Q = ((tan Q)R − S) sec Q, (15) dt d cosec Q = −((ctg Q)R + S) cosec Q, (16) dt

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27

where R=

∞  (−1)k ad2k Q , (2k + 1)! Q

Q ≡

k=0

S=

∞  (−1)k−1 k=1

(2k)!

d Q, dt

(17)

ad2k−1 Q . Q

(18)

Proof. In the paper [6] (P. 964) one obtains the following differentiation formula for the exponent: β −(β−u)H ∂H uH ∂ −βH e = − e du, where H = H(λ) is a linear operator over C, and β, u, and λ are ∂λ ∂λ e 0

complex numbers. Putting β = 1, H = −iQ(t), and λ = t in the above formula, we obtain  1 d iQ e =i ei(1−u)Q Q eiuQ du dt 0 or  1 d iQ iQ e = ie e−iuQ Q eiuQ du. dt 0

(19)

We apply formula (7) to the integrand in the right-hand side of the above equality and then integrate with respect to u. We obtain ∞

 (−i)n d iQ e = ieiQ adn Q , ad0Q Q ≡ Q . dt (n + 1)! Q n=0

We substitute here the exponent eiQ in the form (4) and get 



cos Q + i sin Q = (i cos Q − sin Q)

∞  (−1)n in n=0

(n + 1)!

adnQ Q .

Since ∞  (−1)n in n=0

(n + 1)!

adnQ Q

∞ ∞   (−1)k (−1)k−1 2k−1  2k  adQ Q − i adQ Q ≡ R − iS, = (2k + 1)! (2k)! k=0

k=1

we obtain cos Q + i sin Q

= cos QS − sin QR + i(cos QR + sin QS), which implies equalities (11) and (12). Here the series that define R and S converge in norm, if Q and Q are bounded in this norm. Really, taking into account the definition of these values and properties of norms, we get R ≤ Q  +

1 2 1 2 Q2 Q  + 24 Q4 Q  + · · · = Q (2Q)−1 sinh(2Q), 3! 5!

and 1 1 2Q Q  + Q3 Q  + · · · = Q (2Q)−1 (ch(2Q − 1)). 2! 4! Thus, if Q and Q are bounded, then so are R and S. One can obtain equalities (13)–(16) by the direct differentiation: S ≤

(tan Q) = (sin Q(cos Q)−1 ) = sin Q(cos Q)−1 − sin Q(cos Q)−1 cos Q(cos Q)−1 = (cos QR + sin QS)(cos Q)−1 − tan Q(cos QS − sin QR)(cos Q)−1 = cos QR(cos Q)−1 + sin Q(cos Q)−1 sin QR(cos Q)−1 = (E + (tan Q)2 ) cos QR(cos Q)−1 = (cos Q)−1 R(cos Q)−1 .

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DEREVENSKII

Formulas for derivatives of the cotangent of Q are obtained analogously. The last but one equality takes the form sec Q = −(cos Q)−1 cos Q(cos Q)−1 = −(cos Q)−1 (cos QS − sin QR)(cos Q)−1 = (tan QR − S) sec Q. Formula (16) is obtained analogously. Theorem 2 has the following corollary. Corollary. If in formulas (1)–(3) it holds [Q, Q ] = 0, then equalities (11)–(16) turn into the wellknown formulas for derivatives of scalar trigonometric functions. Formulas (4) and (11) allow us to express derivatives of functions (1) in terms of integrals; in some cases this may appear to be useful. Proposition 4. If Q = Q(t) is a continuously differentiable element of A, then  1 d [cos((1 − u)Q)Q cos(uQ) − sin((1 − u)Q)Q sin(uQ)]du, sin Q = dt 0  1 d cos Q = − [cos((1 − u)Q)Q sin(uQ) + sin((1 − u)Q)Q cos(uQ)]du. dt 0 Proof. If all three exponents in equality (19) take the form (4), then we obtain  1   (cos((1 − u)Q) + i sin((1 − u)Q))Q (cos(uQ) + i sin(uQ))du cos Q + i sin Q = i  =

0 1

0

− cos((1 − u)Q)Q sin(uQ) − sin((1 − u)Q)Q cos(uQ) + i(cos((1 − u)Q)Q cos(uQ) − sin((1 − u)Q)Q sin(uQ)) du.

Comparing real and imaginary parts in this correlation, we obtain two equalities indicated in Proposition 4. The fact that the algebra A is noncommutative significantly complicates the use of its elements. In particular, for this reason one cannot solve problems analogous to those in the scalar theory. Therefore, in order to use well-known methods developed for studying numerical functions in operating with elements of algebras A and B, it is necessary to establish the connection between these elements and their numerical characteristics. This connection is defined by the structure of the vector space inherent in any linear algebra. In this case functional connections between algebraic elements are defined by coefficients of their expansion in the basis of the algebra. In the differentiation of trigonometric functions this is realized as follows. Consider over the algebra A an N -dimensional Lie algebra LN with a commutative multiplication, γ γ (α, β, γ = 1, N ). Note that [Eα , Eβ ] = Cαβ Eγ , where, basic elements Eα , and structure constants Cαβ as everywhere below, the tensor summation is performed over repeating superscripts and subscripts. That is why if Q is defined in the basis of the algebra LN , then commutator series that define R and S are Lie elements. Numerical functions of their projections on Eα represent the main characteristics of the considered functions and their derivatives. Theorem 2. If in functions (1)–(3) it holds Q = q α (t)Eα , where Eα are constant elements of the basis of the algebra LN , and q α (t) are continuously differentiable functions over R, then derivatives of trigonometric functions take form (11)–(16), where R = (q α ) Fαβ Eβ , Fαβ ≡

∞  (−1)k (Dβ )2k , (2k + 1)! α

β Dαβ ≡ q γ Cγα ,

(20)

k=0

RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011

TRIGONOMETRIC SOLUTIONS OF NONLINEAR FIRST-ORDER ODEs

S = (q α ) Hαβ Eβ , Hαβ ≡

∞  (−1)k−1 k=1

(2k)!

(Dαβ )2k−1 ,

29

(21)

and matrix series converge in the norm in MN (the set of N × N -matrices) for any q α such that

 D < ∞, D ≡ Dαβ . Proof. Substituting the expression Q = q α (t)Eα in formula (17), we write R as follows: 1 α 1 ε δ Cβγ Eε + · · · q Eα , [q β Eβ , (q γ ) Eγ ] + · · · = (q α ) Eα − q α q β (q γ ) Cαδ 3! 3!

1 1 ε δ = (q α ) Eα − (q γ ) (q α Cαδ )(q β Cβγ )Eε + · · · = (q α ) δαβ − Dαγ Dγβ + · · · Eβ ≡ (q α ) Fαβ Eβ , 3! 3!

R = (q α ) Eα −

β . where δαβ is the Kronecker delta, i.e., the tensor form of the matrix unit in MN , and Dαβ ≡ q γ Cγα β β β β k k Denoting E = (δα ), D = (Dα ), and (D )α = (Dα ) , we write the latter equality as the representation for R in (20). Formulas (21) are obtained analogously.

The convergence of series that define R and S follows from the fact that each of them consists of terms lesser than the corresponding terms in the series of matrix sines and cosines, which are known to converge in the norm for any bounded values of arguments. Remark. The definition of certain structures in LN automatically simplifies commutation series that define R and S. Thus, if LN is an abelian algebra, then D = 0. In this case, R = (q α ) Eα and S = 0, which reduces formulas (11)–(16) to the usual derivatives of trigonometric functions over a commutative set. If LN is a nilpotent algebra of index r, then the matrix D that defines Q in the associated matrix representation of LN is nilpotent ([4], P. 46). In addition, Dr = 0, which turns series for R and S into polynomials of orders not greater than r − 1. 4. Nonlinear ODE.1 with trigonometric solutions are defined by forms of derivatives of the corresponding functions. Formulas (11)–(16) mean that one can define such equations only if functions over A can be expressed in terms of other ones. Since functions (2) and (3) are defined in terms of sin Q and cos Q which are connected by correlation (8), in order to write nonlinear ODE.1 (NODE.1)√it is necessary that√over A the quadratic equation X 2 = A should have a solution (denoted by X = A). √ √ If, in addition, AB = A B, then since all functions (1)–(3) are commutative, their interconnection obeys the known trigonometric formulas. The parameterization of NODE.1 over A is performed so as to allow one to find arguments of functional solutions from conditions imposed on them by equations. Since in equalities (11)–(16) values R and S are defined by commutation series, we conclude that if Q is defined as an element of the algebra LN , then R and S also belong to this algebra. Consequently, one can parameterize nonlinear differential equations, replacing R and S with some elements A and B of the algebra LN . Then, taking into account formula (8) and the equivalence of equations for cofunctions, equalities (11)–(16) turn into the following three equations:  dX1 = E − X12 A + X1 B, X1 = sin Q, cos Q, (22) dt   dX2 = E + X22 A E + X22 , X2 = tan Q, cot Q, (23) dt  dX3 = X32 − E AX3 + BX3 , X3 = sec Q, cosec Q, (24) dt where (A, B, Xs , Q) ⊂ A, s = 1, 3. Since R and S are interconnected elements of the algebra A, Eqs. (22)–(24) contain only one independent parameter. This means that A and B can be defined parameterically with the help of Q, i.e., with the help of formulas (17) and (18), where R = A and S = B. In addition, the following propositions are valid. RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011

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Proposition 5. If parameters A and B in Eq. (22) take the form A=e

B=

∞  (−1)k ad2k Q , (2k + 1)! Q

(25)

(−1)k−1 2k−1  adQ Q , (2k)!

(26)

k=0 ∞  k=1

where e = ±1 and Q is a continuously differentiable element of A, then its solution equals sin Q with e = 1, and cos Q with e = −1. Proposition 6. If the parameter A in Eq. (23) takes the form (25), then its solution equals tan Q with e = 1, and cot Q with e = −1. Proposition 7. If parameters A and B in Eq. (24) take forms (25) and (26), then its solution equals sec Q with e = 1, and cosec Q with e = −1. However, there is only a theoretical possibility to define parameters of Eqs. (22)–(24) as series (25) and (26), because in practice one usually cannot establish the dependence between A and B with the help of two commutation series. Propositions 5–7 acquire a practical significance only if there exists a realizable algorithm for defining the explicit form of the dependence of one of parameters of NODE.1 on the other one. A Lie algebra over A allows one to find such an algorithm. Let us express A and B in Eqs. (22)–(24) in the basis of the algebra LN : A = aα (t)Eα , B = α b (t)Eα ; then with Q = q α (t)Eα formulas (20) and (21) define the following connection between their projections: ∞  (−1)k α  (Dβ )2k = aβ , (27) (q ) (2k + 1)! α k=0

(q α )

∞  (−1)k−1 k=1

(2k)!

(Dαβ )2k−1 = bβ .

At points of nondegeneracy of the matrix F ≡ (Fαβ ) one can reduce system (27) to the normal form (q α ) = (F −1 )αβ aβ . The class of normalizable systems (27) is nonempty. It contains not only the trivial case of abelian γ = 0) ⇒ (Fαβ = δαβ ) ⇒ (F −1 = F = E), but also more complex cases. The following LN , when (Cαβ proposition gives an example of a globally solvable system. Proposition 8. For nilpotent LN system (27) can always be reduced to the normal form. Proof. As was mentioned above, for a nilpotent Lie algebra the matrix D is nilpotent. This means that MN always contains a nondegenerate matrix T which reduces D and, consequently, F − E to a niltriangular form. Thus, diag(T F T −1 ) = E and (T F T −1 )βα = 0, (α ≤ β) ∨ (α ≥ β). Consequently, det(T F T −1 ) = det F = 1, which makes system (27) normalizable for any q α . The generalization of the structure of LN to a solvable one leads to constraints on t and q α . Thus, even for a two-dimensional nonabelian solvable Lie algebra with the structure of basic elements [E1 , E2 ] = E1 we have ⎞ ⎞ ⎛ ⎛ sin q 2 2 0 0 −q 2 q ⎠ , D2k = (−q 2 )2k−1 D, F = ⎝ ⎠. D=⎝ q1 2 2) 1 1 (q − sin q q 0 (q 2 )2 Consequently, (det F = 0) ⇒ (q 2 = πn). RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011

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In accordance with the Levi theorem ([4], P. 105) an arbitrary Lie algebra is representable as the direct sum of its radical (a maximal solvable ideal) and a semisimple subalgebra. The latter, as follows from the Lie–Engel criterion ([7], P. 223) for the unsolvability of a Lie group, contains simple L3 . Over A there exist two structures of non-isomorphic simple three-dimensional Lie algebras: L3 (VIII) and L3 (IX) in the L. Bianchi classification. Therefore it makes sense to consider the question on the normalization of system (27) for these two particular structures of LN . Proposition 9. For the simple three-dimensional Lie algebra L3 (VIII), possessing the structure of basic elements [E1 , E2 ] = E1 , [E2 , E3 ] = E3 , [E3 , E1 ] = −2E2 ,

(28)

one can reduce system (27) to normal forms: 1) for any q α (q 2 )2 ≤ 4q 1 q 3 , 2) for 0 < (q 2 )2 − 4q 1 q 3 = π 2 n2 , where n is an integer number. γ 1 = C3 = are C12 Proof. According to formula (28), nonzero essential components of the tensor Cαβ 23 2 0.5C13 = 1. This defines the matrix ⎞ ⎛ 2 3 0 ⎟ ⎜−q −2q ⎟ ⎜ 1 . (29) D=⎜ q 0 −q 3 ⎟ ⎠ ⎝ 0 2q 1 q 2

For this matrix we have D 2 = rE + T1 , where r = (q 2 )2 − 4q 1 q 3 , and ⎛ ⎞ 1 3 2 3 3 2 ⎜ 2q q 2q q 2(q ) ⎟ ⎜ ⎟ T1 = ⎜−q 1 q 2 −(q 2 )2 −q 2 q 3 ⎟ . ⎝ ⎠ 1 2 1 2 1 3 2(q ) 2q q 2q q

(30)

Since T12 = −rT1 , we get D2k = r k−1 (rE + T1 ), which allows us to write the matrix ∞ ∞   (−1)k r k (−1)k r k−1 E+ T1 . F = (2k + 1)! (2k + 1)! k=0

(31)

k=1

Consider three cases. 1) If r = 0, i.e., (q 2 )2 = 4q 1 q 3 , then for the corresponding to this variant value F1 = E − 16 T1 we have det F1 = 1. Consequently, F1−1 always exists. 2) For r > 0, when (q 2 )2 > 4q 1 q 3 , we introduce denotations r = ϕ2 and F2 for the given case of the matrix F . Then F2 = ϕ−1 sin ϕE + ϕ−3 (sin ϕ − ϕ)T1 . Substituting matrix (30), we find the determinant det(F2 ) = ϕ−2 sin2 ϕ. Therefore, (det F2 = 0) ⇒ (ϕ = πn), which is indicated in the second condition. 3) Let r < 0, i.e., (q 2 )2 < 4q 1 q 3 . Then denotations r = −f 2 and F3 for the corresponding value of F allow us to write formula (31) in the form F3 = f −1 sinh f E + f −3 (f − sinh f )T1 . The determinant of this matrix equals f −2 sinh2 f ; with f = 0 it differs from zero. This allows us to normalize system (27) at any point. Proposition 10. For the simple three-dimensional Lie algebra L3 (IX), possessing the structure of basic elements [Eα , Eβ ] = εαβγ Eγ ,

(32)

where εαβγ is a completely skew-symmetric unit tensor, one can always reduce system (27) to the normal form, provided that all q α are not concurrently equal to zero. RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011

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Proof. Formula (32) implies that for L3 (IX), ⎛ q3

−q 2

⎞

⎜ 0 ⎟ ⎜ ⎟ 1 D = ⎜−q 3 0 . q ⎟ ⎝ ⎠ q 2 −q 1 0 Consequently, D 2 = −r 2 E + T2 , where r 2 = ⎛

3 

(33)

(q α )2 , and the matrix T2 takes the form

α=1

⎞

(q 1 )2

q1 q2

q1q3

⎜ ⎟ ⎜ ⎟ T2 = ⎜ q 1 q 2 (q 2 )2 q 2 q 3 ⎟ . ⎝ ⎠ 1 3 2 3 3 2 q q q q (q ) In addition, D 4 = (−r 2 E + T2 )2 = r 4 E − 2r 2 T2 + T22 , and since T22 = r 2 T2 , we have D4 = r 2 (r 2 E − T2 ). Using the principle of complete mathematical ∞  r 2k induction, we can prove that D 2k = (−1)k r 2(k−1) (r 2 E − T2 ). This allows us to write F = (2k+1)! E − k=0

∞  k=1

r 2(k−1) (2k+1)! T2 .

Evidently, (r = 0) ⇔ (q α = 0). Therefore, excluding degeneracy points of system (27), where q α = 0, it holds r − sinh r sinh r E+ T2 . (34) F = r r3 The determinant of this matrix det F =

sinh2 r r2

= 0, which makes system (27) normalizable.

Thus, for any algebras, system (27) is normalizable with certain values of the variable t. Therefore, it makes sense to clarify whether it is solvable by quadratures. The Levi theorem on the separation of a radical allows us to answer the stated question in the following three propositions. Proposition 11. In domains, where the left-hand side of system (27) is defined and the system is normalizable, it is integrable by quadratures, provided that LN is solvable. Proof. According to the Lie theorem ([4], P. 64), a solvable algebra LN is a triangular extension of an abelian algebra. Consequently, a similarity transformation reduces the matrix D to a block-triangular form. One of diagonal blocks corresponds to a commutative subalgebra, and the second one does to its triangular extension. Since block-triangular matrices form the subalgebra MN , the normalized system (27) falls into three subsystems. In accordance with summation and multiplication rules for block matrices, the nonlinearities of equations in each of subsystems are formed by the desired functions whose numbers are (in dependence of the form of the triangularity) either greater or less than the number of each of equations. This allows us to sequentially solve three subsystems and, consequently, all the normalized system (27). In the case of semisimple Lie algebras, the question about the solvability of system (27) by quadratures is much more difficult. The next proposition confirms this fact. Proposition 12. For semisimple algebras LN system (27) is not solvable by quadratures with N arbitrary functions aα .

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33

Proof. According to the Lie–Engel theorem mentioned above, a semisimple algebra LN contains one of two simple algebras L3 . Let us show that for both algebras system (27), in general, is not integrable. As was mentioned in the proof of Proposition 9, system (27) is normalizable in three cases. In (q 2 )2 = 4q 1 q 3 . This equality imposes on aα the following constraint: the first case, F1 = E with  2  a2 dt = 4 a1 dt a3 dt. In the second case, with (q 2 )2 > 4q 1 q 3 the matrix F defined by formula (31) takes the form

⎛ ⎞ 1q3 2q3 3 )2 f + 2f q 2f q 2f (q 1 2 2 2 ⎜ ⎟ ⎜ ⎟ 2 2 2 3 , F2 = ⎜ −f2 q 1 q 2 f1 − f2 (q ) −f2 q q ⎟ ⎝ ⎠ 2f2 (q 1 )2 2f2 q 1 q 2 f1 + 2f2 q 1 q 3   πn we where f1 = ϕ−1 sin ϕ and f2 = ϕ−3 (sin ϕ − ϕ). Since det F2 = f12 , with ϕ = (q 2 )2 − 4q 1 q 3 = get ⎛ ⎞ 1 q3 2 q 3 −2(q 3 )2 −2q 2q ⎟ 1 f2 ⎜ ⎜ 1 2 ⎟ 2 )2 2q3 ⎟ . F2−1 = E + ⎜ q q (q q f1 f1 ⎝ ⎠ −2(q 1 )2 −2q 1 q 2 2q 1 q 3 Therefore, system (27) allows the representation ⎧ 1 ⎪ ⎪ (q 1 ) = [a1 (1 + 2f2 q 1 q 3 ) + a2 f2 q 1 q 2 − 2a3 f2 (q 1 )2 ], ⎪ ⎪ f1 ⎪ ⎨ 1 2  (q ) = [−2a1 f2 q 2 q 3 + a2 (1 + f2 (q 2 )2 ) − 2a3 f2 q 1 q 2 ], ⎪ f2 ⎪ ⎪ 1 ⎪ 3  ⎪ ⎩(q ) = [−2a1 f2 (q 3 )2 + a2 f2 q 2 q 3 + a3 (1 + 2f2 q 1 q 3 )]. f1

(35)

Since functions aα are chosen arbitrarily, each of them is a solution to the general Ricatti equation which is not integrable by quadratures. In order to ensure its solvability, it is necessary to impose certain constraints on functional coefficients, for example, those described in [8], [9] (pp. 51–69). f and f2 = f13 (f − In the third case, the normalized system (27) takes the form (35) with f1 = sinh f sinh f ). Consider L3 (IX) with r = 0 ((r = 0) ⇒ (aα = 0)). In this case, F is defined by equality (34), i.e., ⎞ ⎛ 1 )2 1q2 1 q3 + ϕ (q ϕ q ϕ q ϕ 2 2 2 ⎟ ⎜ 1 ⎟ ⎜ 1 2 2 2 2 q3 F = ⎜ ϕ2 q q ⎟, ϕ + ϕ (q ) ϕ q 1 2 2 ⎠ ⎝ ϕ2 q 1 q 3 ϕ2 q 2 q 3 ϕ1 + ϕ2 (q 3 )2 2 where ϕ1 = r −1 sh r and ϕ2 = r −3 (r − sh r). Consequently, F −1 = ϕ11 E − ϕ ϕ1 T2 . With the help of this matrix the normal form of system (27) represents the set of three general Ricatti equations (q α ) = aα (1 − ϕ2 (q α )2 ) − ϕ2 (aα1 q α1 + aα2 q α2 )q α , α = α1,2 = 1, 3, where three coefficients at (q α )s (s = 0, 1, 2) are defined with the help of three arbitrary functions aα . The latter propositions allow us to state the following theorem.

Theorem 3. Assume that in Eqs. (22) and (24), A = aα Eα and B = bα Eα , where (Eα ) is a ∞  (−1)k−1 γ α 2k−1 , and q α are determined from basis of some Lie algebra over A, bα = (q β ) (2k)! (q Cγβ ) k=1

system (27). Then X1 = sin(q α Eα ) is a solution to the first equation, and X3 = sec(q α Eα ) is a solution to the second one. In addition, if LN is solvable, then X1 and X3 are always determined by quadratures in the domain, where the left-hand side of system (27) exists and the system is normalizable; otherwise X1 and X3 are defined only for particular aα . RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011

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DEREVENSKII

Theorem 4. If in Eq. (23), A = aα Eα ∈ LN , then its solution is X2 = tan(q α Eα ), where q α are determined from system (27). In addition, if LN is a solvable algebra, then X2 is always determined by quadratures in the domain, where the left-hand side of system (27) exists and the system is normalizable; otherwise it can be calculated only for particular aα . 5. Example. Assume that A = M2 , N = 2, and L2 = L2 (II), where L2 (II) is a representation in M2 of a two-dimensional nonabelian Lie algebra with the structure [E1 , E2 ] = E1 and basic elements E1 = ( 00 10 ) and E2 = ( 00 01 ). 1. Let us first verify differentiation formulas for trigonometric functions.  1 Evidently, if Q = q α Eα (α = 1, 2), then Q = 00 qq2 . And since Qk = (q 2 )k−1 Q, we have sin q 2 Q, q2

sin Q =

cos Q = E +

cos q 2 − 1 Q. q2

(36)

Differentiating these functions, we obtain q 2 cos q 2 − sin q 2 2  sin q 2  d sin Q = (q ) Q + Q, dt (q 2 )2 q2 1 − q 2 sin q 2 − cos q 2 2  cos q 2 − 1  d cos Q = (q ) Q + Q. dt (q 2 )2 q2

(37) (38)

Let us calculate the same derivatives by formulas (11) and (12). To this end we define R and S by formulas (17) and (18). In accordance with the structure of L2 (II) we have adQ Q ≡ [Q, Q ] = ϕE1 and ϕ = q 1 (q 2 ) − q 2 (q 1 ) , while adnQ Q = (−q 2 )n−1 ϕE1 . Therefore, R = Q + (qϕ2 )2 (q 2 − sin q 2 )E1 and S=

ϕ (1 (q 2 )2

− cos q 2 )E1 , i.e., ⎛ 0 (q 1 ) + R=⎝ 0

Consequently,

ϕ (q 2 (q 2 )2

(q 2 ) ⎛

cos QR + sin QS = ⎝

0 0

⎞ − sin q 2 ) ⎠,

(q 1 )

+

ϕ 2 (q 2 )2 (q

−

⎛ 0 S=⎝ 0

ϕ (1 (q 2 )2

sin q 2 ) +

1 (q 2 ) qq2 (cos q 2

− cos q 2 )

⎞ ⎠.

(39)

0

(q 2 ) cos q 2

⎞ − 1) ⎠.

Hence by equivalent transformations we obtain matrix (37), which confirms Theorem 1.  2  β 0 ) = −q . Since However, when defining R and S by formulas (20) and (21), we get D = (gγ Cγα q1 0 D n = (−q 2 )n−1 D, we have

⎛ sin q 2 2 − sin q 2 q β q2 D=⎝ 1 F = (Fα ) = E + 2 2 q (q ) 2 2 (q 2 )2 (q − sin q ) ⎞ ⎛ cos q 2 −1 2 0 1 − cos q q2 ⎠. D=⎝ 1 H = (Hαβ ) = q (q 2 )2 (1 − cos q 2 ) 0

⎞ 0 ⎠, 1

(q 2 )2

Therefore,

  sin q 2 (q 2 ) q 1 2 2 (q − sin q ) E1 + (q 2 ) E2 , R = (q α ) Fαβ Eβ = (q 1 ) 2 + q (q 2 )2  1   (q ) (q 2 ) q 1 α  β 2 2 + (cos q − 1) + (1 − cos q ) E1 , S = (q ) Hα Eβ = q2 (q 2 )2

(40) (41)

RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011

TRIGONOMETRIC SOLUTIONS OF NONLINEAR FIRST-ORDER ODEs

35

which is equivalent to formulas (39). Consequently, in the considered case the differentiation of the sine function in accordance with Theorems 1 and 2 coincides with the direct differentiation of this function. Differentiation formulas for cos Q can be checked analogously. Consider the function tan Q. In order to define this function, we find cos−1 Q, i.e., the matrix inverse   to the second one in (36). At its nondegeneracy points (det cos Q = cos q 2 = 0) ⇒ q 2 = π2 (2k + 1) we have ⎛ ⎞ 1−cos q 2 1 cos q 2 ⎠ . (42) cos−1 Q = ⎝ −1 2 0 cos q Consequently,

⎛ 0 tan Q = sin Q cos−1 Q = ⎝ 0

q1 q2

⎞ tan q 2

⎠.

(43)

tan q 2

By differentiating this matrix we obtain ⎛ ⎞ (q 1 ) tan q 2 q 1 (q 2 ) (q 2 cos−2 q 2 −tan q 2 0 + d q2 (q 2 )2 ⎠. tan Q = ⎝ dt 2  −2 2 0 (q ) cos q

(44)

The above formula coincides with the product cos−1 QR cos−1 Q, which confirms and illustrates formula (13). Since (sec Q) = − cos−1 Q(cos Q) cos−1 Q, and the validity of the differentiation formula for cos Q needs no verification, we do not comment equalities (15) and (16). 2. Let us illustrate the method for solving nonlinear equations (22)–(24). In order to effectively use the obtained formulas (36)–(44), we apply the given representation of the algebra L2 (II). Below we show that it makes sense to use Theorem 4 first. Let the matrix parameter in Eq. (23) in M2 take the form A = aα (t)Eα . Since L2 (II) is solvable, in accordance with Theorem 4 it has the tangential solution X2 = tan Q, where Q = q α Eα , and q α satisfy 1 = 1, and C γ = 0, (α, β, γ) = (1, 2, 1). Evidently, the desired solution system (27) with N = 2, C12 αβ takes the form (43). System (27) is formed by equating coefficients at E1 and E2 in equality (40) to the functional parameters of the equation, i.e., a1 and a2 , respectively, (q 1 ) sin q 2 (q 2 ) q 1 2 + (q − sin q 2 ) = a1 , q2 (q 2 )2 (q 2 ) = a2 . Since the first equation in this system is a linear ODE.1, it, evidently, is integrable by quadratures:   a1 dt q2 2 , q = a2 dt. (45) q 1 = q 2 ctg 2 1 + cos q 2 Consequently, matrix (43), where q α take the form (45), is a solution to Eq. (23) over M2 with   1 A = 00 aa2 . In order to make sure that it really is a solution, it suffices to substitute X2 in the considered equation. Really, E + X22 coincides with the squared matrix (42), where q 1 and q 2 take the indicated form. Thus, it remains to calculate the product cos−1 QA cos−1 Q, where cos−1 Q is defined in (42), and to compare it with matrix (44). 3. Let us illustrate the way in which Theorem 3 “works”. To this end we need to fix the value of the parameter B in Eq. (22) and to show that matrix (37) with q α in the form (45) coincides with the right-hand side of this equation. RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011

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DEREVENSKII

α  Since  only one of two basic elements of L2 (II) appears in formula (41), in the matrix B = b Eα = 0 b1 , evidently, the element b2 equals zero, and b1 equals the coefficient at E in the right-hand side 1 0 b2   

1 1 of this formula. Therefore, if in Eq. (22) it holds A = 00 aa2 and B = 00 b0 , where b1 = ((q 1 ) q 2 −

a2 q 1 )(q 2 )−2 (cos q 2 − 1), and q α take the form (45), then, according to Theorem 3, its solution is the matrix X1 = sin Q defined in (36). At the same time, verifying identity (8), for formulas (36) we find  2 sin2 q 2 2 (cos q 2 − 1) Q + E + Q sin2 Q + cos2 Q = (q 2 )2 q2 sin2 q 2 2(cos q 2 − 1) (cos q 2 − 1)2 Q + E + Q + Q ≡ E. q2 q2 q2  Therefore, in accordance with the definition of the square root, E − X12 = cos Q. Consequently, substituting formula (37) in the left-hand side of equality (22) and summing the products cos QA and sin QB in the right-hand side, taking into account form (45) of functions q α , we obtain the identity. Really, sin QB = 0 and ⎞ ⎛ a2 q 1 1 2 0 a + q2 (cos q − 1) ⎠. cos QA = ⎝ 2 2 0 a cos q =

Writing (37) in more detail, we get

⎛ 0 d sin Q = ⎝ dt 0

q 2 cos q 2 −sin q 2 2 1 a q (q 2 )2

⎞

+

sin q 2 1  (q ) q2 ⎠

,

a2 cos q 2

which coincides with the previous matrix (accurate to elementary transformations). REFERENCES 1. V. P. Derevenskii, “Hyperbolic Functions and First Order Nonlinear Ordinary Differential Equations over a Banach Algebra,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 6, 7–18 (2006) [Russian Mathematics (Iz. VUZ) 50 (6), 5–16 (2006)]. 2. S. G. Krein (Ed.), Functional Analysis, Ser. “Spravochnaya Matematicheskaya Biblioteka” (Nauka, Moscow, 1972) [in Russian]. 3. V. I. Arnold, Ordinary Differential Equations (Nauka, Moscow, 1971) [in Russian]. 4. N. Jacobson, Lie Algebras (Interscience, New York, 1962; Mir, Moscow, 1964). 5. W. Magnus, “On the Exponential Solution of Differential Equations for a Linear Operator,” Commun. Pure Appl. Math. 7 (4), 649–673 (1954). 6. R. M. Wilcox, “Exponential Operators and Parameter Differentiation in Quantum Physics,” J. Math. Phys. 8 (4), 962–982 (1967). 7. L. P. Eisenhart, Continuous Groups of Transformations (Princeton, Princeton University Press, and London, Oxford University Press, 1933; GIIL, Moscow, 1947). 8. V. P. Derevenskii, “Integrabllity of Riccati Equation and Linear Homogeneous Second-Order Differential Equation,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 33–40 (1987) [Soviet Mathematics (Iz. VUZ) 31 (5), 31–38 (1987)]. 9. V. F. Zaitsev and A. D. Polyanin, Handbook of Ordinary Differential Equations (Fizmatlit, Moscow, 2001) [in Russian].

Translated by O. A. Kashina

RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011