Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/51b/175-remarks-on-integrable-systems_ru.pdf Äàòà èçìåíåíèÿ: Wed Oct 28 19:31:24 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:02:36 2016 Êîäèðîâêà:

ISSN 1560-3547, Regular and Chaotic Dynamics, 2014, Vol. 19, No. 2, pp. 145­161. c Pleiades Publishing, Ltd., 2014.

Remarks on Integrable Systems
Valery V. Kozlov*
Steklov Mathematical Institute, Russian Academy of Sciences ul. Gubkina 8, Moscow, 119991, Russia
Received Septemb er 2, 2013; accepted Septemb er 23, 2013

Abstract--The problem of integrability conditions for systems of differential equations is discussed. Darboux's classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension n. Using a metho d due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields. MSC2010 numbers: 34C14 DOI: 10.1134/S1560354714020014 Keywords: integrability by quadratures, adjoint system, Hamiltonian equations, Euler ­ Jacobi theorem, Lie theorem, symmetries

1. INTRODUCTION We will discuss the problem of integration by quadratures of systems of differential equations. Integration by quadratures implies producing, from a finite numb er of op erations, an algorithm which makes it p ossible to find all solutions to an equation. The allowed op erations include all algebraic op erations on functions, simple quadratures (calculation of the integrals of functions of one variable) and the efficient use of the implicit function theorem. This range of questions is naturally close to differential algebra -- an analog of Galois theory for differential equations (see, e.g., [1]). In connection with this remark, we mention the classical result of Liouville: the linear second-order differential equation x = tx ¨ is not integrable by quadratures. More precisely, its solutions lie in none of the fields which can b e obtained from the field of rational functions of time t by a sequence of finite algebraic extensions, by adding integrals and the exp onentials of integrals. Regarding the methods of exact integration of general nonlinear systems, the fundamental results here are the Euler ­ Jacobi theorem on the integrating factor and the Lie theorem on the solvable algebra of symmetries. The general theory uniting these two approaches is suggested in [2]. Various asp ects of the theory of integration by quadratures are discussed in the b ooks [3, 4]. Are the Euler ­ Jacobi theorem and the Lie theorem a foundation for the classical mechanisms of exact integration of differential equations? The pap er prop oses new constructions which make it p ossible to reduce the problem of exact integration to these standard mechanisms. Our approach is based on two ideas: the interpretation of particular solutions and symmetry fields as first integrals (laws of conservation) of a dynamical system the study of which is useful from the standp oint of integration of the original differential equations.
*

E-mail: kozlov@pran.ru

145


146

KOZLOV

Sections 2 and 3 consider the following problem. Assume that we are given k < n linearly indep endent solutions to the linear system x = A(t)x, x Rn . (1.1) The question is: under what conditions is it p ossible to find all remaining n - k linearly indep endent solutions? It is clear that the answer to this question dep ends, among other things, on the structure of the original system (1.1) and on the prop erties of the known solutions. The differential-algebraic asp ect of this problem is whether the solutions to a given linear differential equation of order n lie in a finite extension of the function field generated by the coefficients and k < n known solutions of this equation. This problem is due to Darb oux who considered the case where n = 3 and the op erator A is skew-symmetric [5]. Darb oux showed that if only one nontrivial (nonzero) solution to the linear equation (1.1) is known, the remaining two linearly indep endent solutions are found by quadratures. Darb oux linked this problem to the Riccati equation and the theory of anharmonic relations. We supplement and develop the classical ideas of Darb oux with particular emphasis on linear Hamiltonian systems. Sections 4 and 5 discuss the general problem of integrability of autonomous systems of differential equations admitting a set of symmetry fields. By an old method due to Liouville we reduce this problem to investigating the integrability of Hamiltonian systems with Hamiltonians linear in the momenta in an extended phase space of dimension that is twice as large. In particular, it turns out that if the original system admits a "complete" Ab elian group of symmetries, its explicit integration can b e derived from the classical Hamilton ­ Jacobi method. New conditions for the integrability of the systems of general differential equations are sp ecified in Section 5. These conditions are based on the application of the conditions for noncommutative integrability of Hamiltonian systems (due to V. A. Steklov). The conditions for the integrability of systems with an "excess" set of nontrivial symmetry fields (their numb er coincides with the dimension of the phase space) are studied in Section 6. For such systems, an integral invariant which plays a substantial role in the problem of exact integration is explicitly sp ecified (it is used in the constructions in Section 7, too). The Liouville technique makes it p ossible to reduce the problem of integration of these systems to the advanced theory of invariant manifolds which are uniquely pro jected onto the configuration space ("general vortex theory" [6]). Dynamical systems on three-dimensional manifolds are dealt with in Section 7. The p ossibility of exact integration of these systems is shown for the case of existence of two symmetry fields which, together with the original vector field, are linearly indep endent at all p oints of the phase space. We emphasize that no additional conditions are imp osed on these fields in this case. In particular, the algebra of symmetry vector fields generated by those fields may b e infinite-dimensional. Finally, Section 8 considers the problem of integrability conditions for nonautonomous systems of differential equations. The role of symmetry fields is played by vector fields frozen into the flow of the original system. They satisfy the Helmholtz equation which is well known in the ideal fluid dynamics. We note that many nontrivial examples of dynamical systems, which can b e used to illustrate the ideas develop ed in the pap er, can b e found in [7­12]. 2. SELF-ADJOINT SYSTEMS We first recall the well-known result ab out the prop erties of the adjoint equation y = -AT (t)y, which is due to Lagrange. If t x(t), t y (t) (2.2) are solutions to equations (1.1) and (2.1), resp ectively, then (y, x) = c = const. Here ( , ) denotes pairing: the value of a covector on a vector (and vice versa).
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 2 2014

yR

n

(2.1)


REMARKS ON INTEGRABLE SYSTEMS

147

Thus, if 1 (t),... ,n (t) is a set of linearly indep endent solutions to the adjoint system, then the solutions to the original system (1.1) will b e found from the linear algebraic relations (1 ,x) = c1 , ... , (n ,x) = cn . Sp ecifically, a linear system is integrated simultaneously with its adjoint system. Theorem 1. Let tr A = 0 (2.3) and assume that n - 1 linearly independent solutions to the system (1.1) are given. Then the remaining solution to the system (1.1) is found by quadratures. It is clear that the adjoint system (2.1) satisfies condition (2.3), too. This condition means that the flow of the system (1.1) (and (2.1)) preserves the standard Leb esgue measure in Rn . Thus, from (2.3) it follows that there exists an additional tensor invariant which is the volume n-form dx1 ... dxn . Proof of Theorem 1. Let 1 (t),... ,
n-1

(t)

b e known linearly indep endent solutions to the system (1.1). Then the adjoint system admits n - 1 functionally indep endent first integrals (y, 1 ),... , (y,
n-1

).

(2.4)

Now we extend the adjoint system (2.1) to the autonomous system of differential equations in the (n + 1)-dimensional space: y = -AT (t)y, t = 1. (2.5)

This dynamical system is nonlinear but (in view of condition (2.3)) its phase flow admits an integral invariant which is the volume (n +1)-form dy1 ... dyn dt. In addition, the dynamical system (2.5) in Rn+1 admits n - 1 indep endent first integrals (2.4). Thus, according to the Euler ­ Jacobi theorem (see, e.g., [13, Chapter 10]), the adjoint system is integrable by quadratures. But then the original system (1.1) is integrable by quadratures, too. As for self-adjoint linear systems (when AT = -A), we may go even further. The following theorem holds. Theorem 2. Let AT (t) = -A(t) for al l values of t, and assume that n - 2 linearly dependent solutions to the system (1.1) are given. Then the remaining two linearly independent solutions to the system (1.1) are found by quadratures. Indeed, self-adjoint systems obviously p ossess the prop erty (2.3). In addition, the system (1.1) has the additional quadratic first integral x2 + ... + x2 . 1 n (2.6)

It remains to make use of the reduction to an autonomous case and reapply the Euler ­ Jacobi theorem. When n = 3, Theorem 2 contains, as a sp ecial case, a classical result due to Darb oux. It seems that Darb oux himself did not link his theorem to the Euler ­ Jacobi theory of the integrating factor.
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 2 2014


148

KOZLOV

The existence of the p ositive-definite quadratic integral (2.6) is a criterion for self-adjointness of the linear system (1.1). Theorem 2 also holds for more general linear systems admitting a timeindep endent first integral in the nondegenerate quadratic form f= 1 (Bx, x), 2 |B | = 0. (2.7)

Actually, it is required to show that the linear system (1.1) is nondivergent in this case, too. Indeed, since (2.7) is a first integral, and the symmetric op erator B is time-indep endent, it follows that BA + AT B = 0. Hence, AT = -BAB Therefore tr A = tr AT = - tr BAB
-1 -1

(2.8)

.
-1

= - tr AB

B = - tr A,

whence we obtain tr A = 0. Q.E.D. In this case the system (1.1) will b e self-adjoint with resp ect to the metric in Rn (generally pseudo-Euclidean) which is given by the quadratic form (2.7). We show that the nondegenerate linear system (1.1) (|A(t)| = 0) can b e represented in the "Hamiltonian" form = x f , x (2.9)

where = BA-1 is a nondegenerate time-dep endent skew-symmetric op erator, and the "Hamiltonian" f is given by the formula (2.7). The skew-symmetry of the op erator which follows from the equality (2.8) is not obvious here. Indeed, from (2.8) we have: B Thus, T = (AT )-1 B = - BA-1 B
-1 -1

(AT )-1 = -A-1 B

-1

. B = - .

The symplectic structure is given by the nondegenerate skew-symmetric matrix which is timedep endent in the general case. This is why the system (2.9) will b e Hamiltonian in the usual sense if the original linear system (1.1) is autonomous. From (2.9) it follows (as in the Hamiltonian case) that f is a first integral. Indeed, multiplying b oth parts of (2.9) by x, we obtain f ,x x = (x, x) = 0.

3. HAMILTONIAN SYSTEMS We now consider the linear Hamiltonian system H H , y=- , x Rn , y Rn , x= y x 1 1 H = (Ax, x)+ (Bx, y )+ (Cy , y ), 2 2 the op erators A and C are symmetric; A, B and C dep end, in a certain way, on t. Let z= x, y W= A B, BT C J= 0 En . -En 0
Vol. 19 No. 2 2014

(3.1)

REGULAR AND CHAOTIC DYNAMICS


REMARKS ON INTEGRABLE SYSTEMS

149

Here, En is an n-order identity matrix. Then the system (3.1) can b e represented in a more compact form as: 1 H , H = (Wz , z ). (3.2) J z = -Wz = - z 2 Since J 2 = -E2n , z = JW z . The adjoint system has the form u = WJ u. If we set v = Ju, the adjoint system (3.3) takes the original Hamiltonian form: J v = -Wv . Thus, according to Lagrange, if u(t) is a solution to the system (3.3), the Hamiltonian system (3.2) admits the linear integral (u(t),z ) = (v (t),J z ) = const. In the original canonical variables, we obtain the following statement of Poincar´ if x = a(t), e: y = b(t) are a solution to the linear Hamiltonian equations (3.1), then this system admits the linear first integral f = (y, a(t)) - (x, b(t)). Incidentally, the system of equations (1.1) and (2.1) can b e represented in the canonical form of (3.1) if we set H = (Ax, y ). Let a1 (t) and b1 (t) b e another solution to Eqs. (3.1). Then these equations admit one more linear integral f1 = (y, a1 (t)) - (x, b1 (t)). Their Poisson bracket {f, f1 } = equals the constant (a, b1 ) - (a1 ,b). Theorem 3. Assume that n linearly independent solutions to the Hamiltonian system (3.1) x = ai (t), are given and al l numbers ij = (ai ,bj ) - (aj ,bi ) (3.4) y = bi (t), 1 i n f f1 f f1 - yi xi xi yi (3.3)

are equal to zero. Then the linear system of differential equations (3.1) is integrable by quadratures. This statement is a simple corollary from the classical Liouville theorem on the complete integrability of Hamiltonian systems: n linear functions (y, ai ) - (x, bi ) are functionally indep endent first integrals which are pairwise in involution. An explicit integration of the linear equations (3.1) under the conditions of Theorem 3 (as well as the the proof of the Liouville theorem) can b e p erformed by the Hamilton ­ Jacobi method. As
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 2 2014


150

KOZLOV

an illustration, consider an autonomous Hamiltonian system with one degree of freedom (n = 1) and the Hamiltonian H= x2 y2 + 2 . 2 2

Here, is a nonvanishing time function. Assume that we are given a solution to this system: y = a(t), The functions a and b satisfy the relations b = a, a = - 2 b. x = b(t). (3.5)

The Hamiltonian equations admit the first integral yb - xa = c (= const). Assume that b = 0. Then y= c a x+ . b b (3.6)

According to the Hamilton ­ Jacobi method, one should find the function S (t, x, c) (c is a parameter) which satisfies the relations S = y, x S = -H, t

where the momentum y and the Hamiltonian H are represented as the functions x and t using (3.6). Omitting the elementary calculations, we obtain S= c2 a2 c x + x- 2b b 2 b2 dt . (t)

By the Jacobi theorem, the coordinate x is found from the relation S = d, c where d is a constant. Hence, x = db + bc b2 dt , (t) y= c + da + ac b b2 dt . (t)

The coefficients for the new parameter d yield an a priori known solution to the Hamiltonian equations, therefore the new solution can b e represented by the formulas x(t) = b dt , b2 y (t) = 1 +a b dt . b2

These formulas can also b e given a meaning when, at a certain instant t = , the function b(t) vanishes. Resolving the indeterminacy using de L'Hospital's rule, we have:
t

lim x(t) = -

1 , a( )

t

lim y (t) = 0.

It is clear that a( ) = 0, otherwise the solution (3.5) would degenerate into a trivial equilibrium.
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 2 2014


REMARKS ON INTEGRABLE SYSTEMS

151

Theorem 3 can b e generalized. Assume that we are given k n linearly indep endent solutions x = ai (t), y = bi (t) (1 i k) to the linear Hamiltonian system (3.1). Introduce the matrix k â k =
ij

,

comp osed of the numb ers (3.4). This is a skew-symmetric matrix, therefore its rank is an even numb er; we denote it by 2s. The inequality n+s k is well known in symplectic geometry. Theorem 4. If n + s = k, then the linear Hamiltonian equations (3.1) are integrable by quadratures. When s = 0 (k = n), we obtain Theorem 3. Equality (3.7) is known as a condition for noncommutative integrability of Hamiltonian systems: a Hamiltonian system admits k n functionally indep endent first integrals, though not all of them are pairwise in involution. Usually, autonomous Hamiltonian systems are considered. However, any nonautonomous Hamiltonian system can b e made autonomous by increasing the numb er of degrees of freedom by 1. In this case the numb er of first integrals is also increased by 1 without an increase in the matrix rank of their Poisson brackets. The essential condition for the applicability of noncommutative integration theory is that the Poisson bracket of any pair of integrals is a function of all first integrals. In our case, this condition is automatically met since all Poisson brackets are constants in the whole phase space. It is customary to link the theory of noncommutative integration of Hamiltonian systems to the works [14] and [15], where the emphasis is placed on studying the top ology of common level sets of all first integrals and the structure of their phase flows. Integrability by quadratures was later established by A. V. Brailov [16] by reducing to the well-known Lie theorem on the integrability of dynamical systems with a complete Ab elian symmetry group. Actually, general ideas of noncommutative integration theory were develop ed in the works by S. Lie (and later by E. Cartan), and explicitly formulated by V. A. Steklov in the short note [17]. A historical commentary can b e found in [18], where an alternative extension of the classical Hamilton ­ Jacobi method is presented. The method may b e useful for explicit integration of linear Hamiltonian equations under the conditions of Theorem 4. 4. INTEGRATION OF SYSTEMS WITH ABELIAN SYMMETRY GROUP AND THE HAMILTON ­ JACOBI METHOD It is customary to prove the Liouville theorem on integrability by quadratures of Hamiltonian systems with a complete set of involutive integrals is proved by the Hamilton ­ Jacobi method (see, e.g., [13]). However, this can also b e done by applying the well-known Lie theorem to the restriction of Hamiltonian systems to the manifolds of the level set of known first integrals. A more general theorem on the noncommutative integration of Hamiltonian systems (see [16]) is proved in a similar way. We recall the formulation of the Lie theorem confining ourselves to consideration of its local asp ect. Let there b e given n vector fields v1 ,... ,vn in a domain Rn = {x} and assume that these vector fields are · linearly indep endent at each p oint of the domain, · commute pairwise with each other (i.e., [vi ,vj ] = 0 for all 1 Then each of the differential equations x = vi (x) is integrable by quadratures. It turns out that the Lie theorem itself can b e proved by the Hamilton ­ Jacobi method. At first glance, it seems surprising (and even paradoxical), since nothing is said ab out symplectic structures and Hamiltonian equations in the formulation of the Lie theorem.
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 2 2014

(3.7)

i, j

n).


152

KOZLOV

Our proof uses an old observation by Liouville: any system of differential equations may b e always considered as a half of some Hamiltonian system in a double-dimensional phase space. Indeed, consider the general dynamical system x = v (x), x M. (4.1) We introduce a space of the cotangent bundle = T M with a standard symplectic structure (the elements Tx M are covectors y at the p oint x) and the Hamiltonian function H = (y, v (x)), (4.2) which is the value of the covector y on the vector v (and vice versa). The first of the differential Hamiltonian equations x= H , y y=- H x

coincides with (4.1), and the second equation y=- v x
T

y

is a variational equation conjugate to (4.1). According to Lagrange's general observation, pairing (4.2) is a first integral of Hamiltonian equations (which is, of course, clear anyway). Now to the vector fields v1 ,... ,vn we assign the linear functions in momenta F1 = H = (y, v1 ), ... , Fn = (y, vn ). The following equality for the Poisson brackets is easily verified. {Fi ,Fj } = (y, [vi ,vj ]). Hence, the functions (4.3) are in involution and functionally indep endent: (F1 ,... ,Fn ) =0 (y1 ,... ,yn ) (4.4) (4.3)

in view of the assumption of linear indep endence of the vectors v1 ,... ,vn . After that we can use a proof of the Liouville theorem which relies up on the Hamilton ­ Jacobi method. In order to find a complete integral of the Hamilton ­ Jacobi equation S S +H = + t t S ,v1 x = 0, (4.5)

one needs to equate the functions (4.3) to arbitrary constants (y, v1 ) = c1 , ... , (y, vn ) = cn , express the momenta y as functions of x and c, and consider the differential 1-form yi dxi - Hdt. (4.6)

The circumflex accent denotes the result of the substitution; it is clear that H = c1 . In view of the involution prop erties of the functions (4.3), the 1-form (4.6) is closed: locally, it is a total differential of some function of x and t (and of the parameters c = (c1 ,... ,cn )) S (x, t, c). (4.7) n simple quadratures are sufficient to find it. The function (4.7) is a complete integral of the Hamilton ­ Jacobi equation (4.5), since F 2S = , x c y which is nonzero according to (4.4).
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 2 2014

(4.8)


REMARKS ON INTEGRABLE SYSTEMS

153

The function (4.7) is S= where s1 ,... ,sn ar of the vector fields (i.e., the solutions following algebraic sj (x)cj - c1 t,

e expressed by algebraic op erations and quadratures in terms of the comp onents v1 ,... ,vn . Therefore, by the Jacobi theorem, the coordinates x as functions of t of the original system of differential equations x = v1 (x)) are found from the system s1 (x) - t = d1 , s2 (x) = d2 ,... ,sn (x) = dn , (4.9)

where d1 ,... ,dn are arbitrary constants. According to (4.8), this system may b e transformed and the coordinates x may b e represented as functions of time and arbitrary constants d1 ,... ,dn . Q.E.D. It is useful to compare this proof to the original proof of S. Lie (see, e.g., [3, 19]). The original proof differs from our proof only in form. In any case, Lie also derived the equalities (4.9) though by a different method. Note that the change of the variables z1 = s1 (x1 ,... ,xn ), ... , zn = sn (x1 ,... ,xn ) rectifies the tra jectories: the equations in the new variables take a particularly simple form z1 = 1, z2 = ... = zn = 0. original works, also used the theory of Hamiltonian facts which at first glance are not related to it. It is for his own proof of the fundamental theorem which to any abstract Lie algebra.

Note, incidentally, that Sophus Lie, in his systems and Poisson brackets to prove some the idea of "Hamiltonization" that is a basis states that some local Lie group corresp onds

5. INTEGRABILITY CONDITIONS FOR DIFFERENTIAL EQUATIONS WITH A REDUNDANT NONCOMMUTATIVE GROUP OF SYMMETRIES The Lie theorem may b e generalized in the way suggested by its proof given in Section 4. Assume that we are given the autonomous system of differential equations x = u(x), x M. (5.1) Phase flows of the vector fields commuting with the field u are one-parameter groups of symmetries: each such transformation maps a solution to the system (5.1) into a solution to the same system. We consider the set of vector fields v1 (= u),v2 ,... ,vm , which generates the Lie algebra G with resp ect to the commutation op eration: [vi ,vj ] = ck vk , ij ck = const. ij (5.2)

We assume that the fields v2 ,... ,vm are fields of symmetries for the system (5.1): [v1 ,vi ] = 0, i 1. We now define the dimension (dim G ) and rank (rank G ) of the algebra G . First we associate with the vector fields (5.2) the set of functions linear in momenta F1 = (y, v1 ), ... , Fm = (y, vm ), (5.3) which are defined in the extended phase space = T M . Linear combinations of these functions generate a finite-dimensional Lie algebra with resp ect to a Poisson bracket, which is obviously isomorphic to the algebra G . By a dimension of the algebra G we mean the maximal rank of the Jacobian matrix (F1 ,... ,Fm ) . (y1 ,... ,yn ,x1 ,... ,xn )
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 2 2014


154

KOZLOV

Generally sp eaking, the rank of this matrix dep ends on a p oint in the extended phase space and is maximal almost everywhere. Without loss of generality, we assume that dim G = m. (5.4) In other words, the functions (5.3) form a functionally indep endent set. The dimension defined in this way may b e called functional. An "algebraic" dimension of the algebra G could b e introduced as a maximum numb er of its linearly indep endent elements (5.3). This value is equal to the dimension of the space of linear combinations of the vector fields (5.2) (rememb er that linearly indep endent vector fields can b e linearly dep endent at all p oints of M ). An algebraic dimension is always not less than a functional dimension; but these numb ers may not coincide. The rank of the algebra G is a maximal rank of the skew-symmetric matrix of Poisson brackets for the functions (5.3): rank G = max rank {Fi ,Fj } .
x,y

The brackets {Fi ,Fj } are linearly expressed in terms of values of the functions (5.3). This rank is maximal for almost all values F1 ,... ,Fm . The rank is always even; we set rank G = 2s. Theorem 5. If 1 rank G , 2 then the differential equation (5.1) is integrable by quadratures. dim G = dim M + (5.5)

If (5.4) holds, then condition (5.5) takes the form m = n + s. Theorem 5 is easily derived from the theory of noncommutative integration of Hamiltonian systems with the use of the construction from Section 4. If the algebra of symmetry vector fields is Ab elian (rank G = 0), then Theorem 5 b ecomes a classical Lie theorem. As a simple illustration, assume that M = R3 with Cartesian coordinates x1 ,x2 ,x3 . We introduce four vector fields u = v1 ,v2 ,v3 ,v4 , which are defined by the differential op erators L1 = x1 g + x2 g + x3 g , L2 = x3 - x2 , x1 x2 x3 x2 x3 - x3 , L4 = x2 - x1 . L3 = x1 x3 x1 x1 x2

(5.6)

Here, g is an arbitrary nonzero function of distance (x2 + x2 + x2 )1/2 . Note that the vector fields 1 2 3 v2 ,v3 ,v4 are linearly dep endent at each p oint, but none of their nontrivial linear combinations with constant coefficients yields a null equation. It is clear that [v1 ,vj ] = 0 (j 1), [v2 ,v3 ] = v4 , [v3 ,v4 ] = v2 , [v4 ,v2 ] = v3 . It is easy to verify that dim G = 4, and rank G = 2. Consequently, condition (5.5) holds, which implies the integrability by quadratures of the equation x = gx, x R3 . That, however, was obvious from the very b eginning. We emphasize that by Theorem 5 integration of the system of differential equations (5.1) given on the n-dimensional manifold M is actually p erformed in the 2n-dimensional extended phase space = T M . We show how to do it using the example considered ab ove. We associate with the vector fields (5.6) the four functions F1 = g xi yi , F2 = x3 y2 - x2 y3 , F3 = x1 y3 - x3 y1 , F4 = x2 y1 - x1 y2 .

It is clear that the three functions
2 2 2 F1 , F2 and F = F2 + F3 + F4

REGULAR AND CHAOTIC DYNAMICS

Vol. 19

No. 2

2014


REMARKS ON INTEGRABLE SYSTEMS

155

are indep endent in a six-dimensional phase space and are pairwise in involution. Thus, by the Liouville theorem, Hamiltonian differential equations with the Hamiltonian F1 are integrable by quadratures. But the vector field on M = R3 cannot b e associated with the function F since the function is quadratic in momenta. We complement Theorem 5 with one statement derived from the Euler ­ Jacobi theorem on the integrating factor. Again, we consider the set of vector fields (5.2) with the commutation condition [v1 ,vj ] = 0, 1 j m.

But in this case the fields are not assumed to generate the Lie algebra. We associate with them the functions (5.3) linear in momenta and consider them to b e functionally indep endent: rank (F1 ,... ,Fm ) = m. (x, y )

Theorem 6. If m = 2n - 2, then the system of differential equations (5.1) is integrable by quadratures. If n = 2, then m = 2. In this case, Theorem 6 b ecomes a classical Lie theorem. If n = 3, then in order to integrate by quadratures, it is sufficient to have three "nontrivial" symmetry fields (as in the example (5.6), but these fields do not necessarily generate the three-dimensional Lie algebra). Theorem 6 is proved very simply. The Hamiltonian system with the Hamiltonian F1 = (y, u) admits m = 2n - 2 functionally indep endent first integrals F1 ,... ,Fm and its phase flow preserves the standard volume form in the 2n-dimensional phase space. 6. SYSTEMS WITH A REDUNDANT SET OF SYMMETRIES AND GENERAL VORTEX THEORY We again assume that M is an n-dimensional manifold with the local coordinates x1 ,... ,xn . Assume that the dynamical system x = u(x), admits n symmetry fields v1 ,... ,vn , (6.2) which are linearly indep endent at each p oint x M : [u, vi ] = 0 for all i = 1,... ,n. The field u itself generates a trivial group of symmetries, which is a family of shifts along the tra jectories of the system (6.1). Thus, we consider the simplest variant of the system with "redundant" symmetries. Since the vectors (6.2) are linearly indep endent at all p oints, we have the single-valued representation u= i vi , (6.3) xM (6.1)

where 1 ,... ,n are smooth functions of x. We also introduce the volume element (measure) d on M and set d = |V |-1 dx1 ... dxn , (6.4) where the columns (or rows) of the n â n matrix V are comp osed of the comp onents of the vectors (6.2). Theorem 7. The functions 1 ,... ,n from (6.3) are constant on the trajectories of the system (6.1), while (6.4) is an integral invariant of this system. It is clear that the invariant measure (6.4) is correctly defined in the region of the phase space M , where the vectors (6.2) are linearly indep endent. As for the functions 1 ,... ,n ,
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 2 2014

(6.5)


156

KOZLOV

some (or even all) of them may b e constant. In this case th integrating the original system (6.1). In any case, if the field functions (6.5) there can b e only n - 1 indep endent ones. We give an example showing the p ossibility that there indep endent coefficients from the set (6.5). Let M = Rn = {x1 is given by the differential op erator . x1 Assume that the differential op erators e-
x
2

ey are useless for the purp ose of u has no zeros, then among the exist exactly n - 1 functionally ,... ,xn }; assume that the field u (6.6)

+ e- x1

x 2 -x

3

, x2

-e-
2x
n

2x

3

+ e- x2
-1

x 3 -x

4

-e-

xn

+ e-

x n -x

2

, -e-2x4 + e- x3 x3 , -e-2x2 xn xn

x 4 -x

5

, x4

... ,

corresp ond to the fields (6.2). When n > 1, all these op erators commute with the op erator (6.6) and, therefore, generate symmetry fields. Here 1 = ex2 , 2 = ex3 ,... ,n
-1

= exn , n = ex2 .

The first n - 1 of them are obviously functionally indep endent. We shall prove Theorem 7 using general vortex theory [6], which studies the geometry and dynamics of invariant manifolds of the Hamiltonian equations uniquely pro jected onto configuration space. As in Sections 4 and 5, we again introduce the extended phase space = T M with a natural symplectic structure, and associate the functions on linear in momenta H = (y, u), F1 = (y, v1 ), ... , Fn = (y, vn )

with the field u and the vector fields (6.2). Since the field u commutes with the vector fields (6.2), {H, Fi } = 0 for all 1 i n. Therefore, the functions F1 ,... ,Fn are first integrals of the Hamiltonian system with the Hamiltonian H . In particular, the integral surfaces {x, y : F1 = c1 ,... ,Fn = cn } (6.7) will b e invariant manifolds of the Hamiltonian equations with the Hamiltonian function H . Since the vector fields (6.2) are linearly indep endent at all p oints of M , the integral surfaces (6.7) are smooth n-dimensional manifolds which are uniquely pro jected onto M . Expressing the momenta in terms of coordinates, we obtain the field of momenta on M y = p(x, c), x M, which linearly dep ends on the parameters c1 ,... ,cn . This field satisfies the Lamb equation [6]: p - x p x
T

u=-

h , x

(6.8)

where h = (p(x, c),u) is a restriction of the Hamiltonian to the invariant manifold (6.7). According to the general theory, the vector field H y

y =p

should app ear instead of the field u in the Lamb equation. But this vector field does not dep end on c and exactly coincides with the field u. It immediately follows from (6.8) that the function h is constant on the tra jectories of the original dynamical system (6.1). It is easy to understand that h= i (x)ci .
Vol. 19 No. 2 2014

REGULAR AND CHAOTIC DYNAMICS


REMARKS ON INTEGRABLE SYSTEMS

157

Since the parameters c1 ,... ,cn can indep endently take arbitrary values, it follows that the first part of the conclusion of Theorem 7 holds. The invariance of measure (6.4) follows from the general result ab out the invariance of the measure with the density = p =V c
-1

= |V |-

1

(see [6, Chapter I I]). Note, incidentally, that the constancy of the functions (6.5) on the tra jectories of the system (6.1) can also b e proved by a straightforward verification. Indeed, from (6.3) we obtain: 0 = [u, u] = where i = Lu i = i (j u xj
)

[u, i vi ] =

i [u, vi ]+

i vi =

i vi ,

(6.9)

is a total derivative of the function i by virtue of the system (6.1). Since the vector fields (6.2) are linearly indep endent at all p oints of M , 1 = ... = n = 0. Q.E.D. From Theorem 7 (taking into account the Euler ­ Jacobi theorem) it follows that Corollary 1. If among n functions there are n - 2 functional ly independent ones, then the differential equations (6.1) are integrable by quadratures. 7. SYSTEMS ON THREE-DIMENSIONAL MANIFOLDS For systems with low-dimensional phase space, one can p oint out simple and practically final conditions for exact integration. We consider the dynamical system x = u(x), xM (7.1) without equilibrium p ositions. For simplicity, we assume that all ob jects are analytic. In particular, the connected manifold M has an analytic structure, and the comp onents of the vector field u (and of symmetry fields) analytically dep end on local coordinates on M . Theorem 8. Let dim M = 3 and assume that two symmetry fields v2 and v3 ([u, vj ] = 0, j = 2, 3) are given, and the vectors u = v1 (x), v2 (x) and v3 (x) (7.2) are linearly independent at least at one point of M (therefore, they are independent almost everywhere). Then almost al l solutions to the differential equation (7.1) are found by quadratures. We emphasize that (unlike the Lie theorem) we imp ose no additional conditions on the vector symmetry fields v2 and v3 . A priori, the algebra of vector symmetry fields can also b e infinitedimensional. Proof of Theorem 8. Let U b e an op en domain in M , where the vectors (7.2) are linearly indep endent. In view of analyticity, the set M \ U has zero measure. In the domain U the dynamical system (7.1) admits an integral invariant of the form (6.4). This follows from Theorem 7 applied to the vector fields v1 = u, v2 , v3 , which are linearly indep endent at each p oint of the domain U . We now decomp ose the commutator [v2 ,v3 ] = w into the linearly indep endent vectors (7.2) as in the basis: w = 1 v1 + 2 v2 + 3 v3 . [u, [v2 ,v3 ]] + [v2 , [v3 ,u]] + [v3 , [u, v2 ]] = 0,
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 2 2014

(7.3)

The coefficients 1 , 2 , 3 are analytic functions in the domain U . According to the Jacobi identity


158

KOZLOV

the commutator [v2 ,v3 ] is also a symmetry field: [u, w] = 0. From (7.3) we obtain: [v1 ,w] = i vi = 0

(compare to (6.9)). But then 1 , 2 and 3 are first integrals of the dynamical system (7.1). If not all of them are constant, then the integration by quadratures of Eq. (7.1) follows from the Euler ­ Jacobi theorem on the integrating factor. It remains to consider the case where all decomp osition coefficients (7.3) are constant. We show that in this case the linear combinations of the vector fields (7.2) generate the three-dimensional solvable Lie algebra G with resp ect to the commutation op eration. Let us use the commutation relations [w, v1 ] = 0, and consider three cases: 1. 1 = 2 = 3 = 0, 2. 2 + 2 = 0, 2 3 3. 1 = 0, 2 = 3 = 0. In the first case the algebra of vector fields G is commutative (and hence solvable). Consider the second case and let, for instance, 2 = 0. Then v2 = and the basis in G is comp osed of the fields w, v3 , v1 . According to (7.4), we have the chain of emb edded ideals {w} {w; v3 } {w; v3 ; v1 }, which means that the algebra G is solvable. In the third case the basis in G is comp osed of the fields w = 1 v1 , v2 , v3 . The solvability of the algebra G follows from the sequence of emb edded ideals: {w} {w; v2 } {w; v2 ; v3 } (see (7.4)). Integrability by quadratures of the original system (7.1) in Lie theorem, and the integrability in the second case follows Lie theorem established in [20]. This strengthening implies th and x = v1 (x) are integrable by quadratures, along with the cases 1 and 3 follows from the classical from the strengthened version of the at the differential equations x = v3 (x) differential equation x = w(x)
Vol. 19 No. 2 2014

[w, v2 ] = -3 w,

[w, v3 ] = 2 w

(7.4)

1 3 w- v3 2 2

REGULAR AND CHAOTIC DYNAMICS


REMARKS ON INTEGRABLE SYSTEMS

159

8. FROZEN FIELDS AND INTEGRATION OF NONAUTONOMOUS SYSTEMS OF DIFFERENTIAL EQUATIONS The observations from Sections 4 and 5 can b e carried over to a nonautonomous case. Thus, we consider the nonautonomous system of differential equations x = u(x, t), x M n. (8.1) (8.2) The role of symmetry fields will b e played by the vector fields v1 (x, t),... ,vm (x, t), which are frozen into the flow of the system (8.1). They satisfy the following relations: vi +[vi ,u] = 0, 1 i m. (8.3) t Here, the commutator is calculated at fixed t. These relations can b e represented in the equivalent form: v + Lu v = 0, (8.4) t where Lu is the Lie derivative along the vector field u. Equation (8.4) expresses the fact that the flow of the system (8.1) "transfers" the vector field v . Equation (8.3) is known in hydrodynamics as the Helmholtz equation. In particular, the vortex vector of barotropic flows of an ideal incompressible fluid satisfies this equation. The classical Helmholtz theorem on the vortex lines b eing frozen-in is derived from (8.3). Theorem 9. Let m = n and assume that at each instant of time the vector fields (8.2) are linearly independent at al l points x M , commute with each other ([vi ,vj ] = 0) and satisfy Eqs. (8.3). Then the original system (8.1) is integrable by quadratures. This statement is a variation on the theme of the Lie theorem on the systems with a complete Ab elian symmetry group. First we show how to reduce it to an autonomous case and then derive it from the Liouville theorem on completely integrable Hamiltonian systems. We extend the phase space M to an (n + 1)-dimensional space by adding t as an indep endent variable. Then the system (8.1) turns into an autonomous one: x = u(x, t), t = 1. (8.5) We extend the vector fields vi in a different manner, namely, by associating with them the differential equations x = vi (x, t), t = 0. (8.6) Let u (vi ) b e a vector field in M â Rt , which is defined by the right-hand side of (8.5) (resp ectively (8.6)). It is clear that the relations (8.3) are equivalent to the commutation conditions [u, vi ] = 0, 1 i n, and the assumption on the commutativity of the vector fields v1 ,... ,vn for fixed t means the commutativity of v1 ,... , vn . It remains to note that the vectors u, v1 ,... , vn are linearly indep endent at all p oints of the extended phase space. We now give the "Hamiltonian" version of the proof of Theorem 9. On the extended phase space = T M we introduce the Hamiltonian H = (y, u(x, t)) and n functions F1 = (y, v1 (x, t)),... ,Fn = (y, vn (x, t)). It is clear that by virtue of the Hamiltonian system the total derivatives of these functions Fi + {H, Fi } Fi = t
REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 2 2014

(8.7)


160

KOZLOV

are equal to zero in view of the relations (8.3). Then, {Fi ,Fj } = 0 and the functions (8.7) are functionally indep endent on for all t. Consequently (according to the Liouville theorem [13]), Hamiltonian equations with Hamiltonian H are found by quadratures. But then the closed system of differential equations x= is integrable Theorem illustration, equation on H = u(x, t) y

by quadratures, too. 9 gives interesting explicit formulas for the solution of nonautonomous systems. As an we consider the simplest case where n = 1. Thus, consider a nonautonomous differential the straight line x = u(x, t), x R. (8.8)

Let v (x, t) = 0 b e another vector field satisfying Eq. (8.3): v u v + u= v. t x x The solution of (8.8) is given by the formula
x t

(8.9)

dp - v (p, t)
0 0

u(0,s) ds = , v (0,s)

(8.10)

where is an arbitrary constant. This formula can b e verified by differentiation using (8.9). But we shall give its derivation using the Hamilton ­ Jacobi method. The Hamilton ­ Jacobi equation is S S + u = 0. (8.11) t x We set F = y · v (x, t). It is a first integral in view of (8.9). Setting F = c, we find the momentum c S =. x v According to (8.11) and (8.12) the op eration S satisfies the following system y= c S S =, = -c x v t This system is consistent in view of the assumption (8.9). up to an irrelevant additive constant,
x t

(8.12)

u . v Integrating these equations, we obtain,

S=c
0

dp -c v (p, t)
0

u(0,s) ds. v (0,s)

By the Jacobi theorem, S = = const, c whence Eq. (8.10) follow Let, for instance, the solution v = (x). After separable variables. The ab ove allows us nonautonomous case. s. variables in (8.8) b e separable: u = (t)(x). Then (8.9) has an obvious that (8.10) turns into a classical formula used to solve equations with to easily formulate and prove the analogues of Theorems 5 and 6 for a

REGULAR AND CHAOTIC DYNAMICS

Vol. 19

No. 2

2014


REMARKS ON INTEGRABLE SYSTEMS

161

9. ACKNOWLEDGMENTS The author is grateful to the referee for helpful discussions and comments. REFERENCES
1. Kaplansky, I., An Introduction to Differential Algebra, Paris: Hermann, 1957. 2. Kozlov, V. V., The Euler ­ Jacobi ­ Lie Integrability Theorem, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 329­343. 3. Olver, P. J., Applications of Lie Groups to Differential Equations, Grad. Texts in Math., vol. 107, New York: Springer, 1986. 4. Borisov, A. V. and Mamaev, I. S., Modern Methods of the Theory of Integrable Systems, Moscow: R&C Dynamics, ICS, 2003 (Russian). 5. Darboux, G., Le¸ s sur la th´ con eorie g´ ´ ale des surfaces et les applications g´ ´ riques du calcul ener eomet infinit´ esimal: T. 1, Paris: Gauthier-Villars, 1914. 6. Kozlov, V. V., General Theory of Vortices, Encyclopaedia Math. Sci., vol. 67, Berlin: Springer, 2003. 7. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Lie algebras in vortex dynamics and celestial mechanics: 4, Regul. Chaotic Dyn., 1999, vol. 4, no. 1, pp. 23­50. 8. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Multiparticle Systems. The Algebra of Integrals and Integrable Cases, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 18­41. 9. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Superintegrable system on a sphere with the integral of higher degree, Regul. Chaotic Dyn., 2009, vol. 14, no. 6, pp. 615­620. 10. Borisov, A. V. and Mamaev, I. S., Superintegrable systems on a sphere, Regul. Chaotic Dyn., 2005, vol. 10, no. 3, pp. 257­266. 11. Borisov, A. V. and Mamaev, I. S., On the problem of motion of vortex sources on a plane, Regul. Chaotic Dyn., 2006, vol. 11, no. 4, pp. 455­466. 12. Borisov, A. V. and Pavlov, A. E., Dynamics and statics of vortices on a plane and a sphere: 1, Regul. Chaotic Dyn., 1998, vol. 3, no. 1, pp. 28­38. 13. Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., New York: Cambridge Univ. Press, 1959. 14. Nekhoroshev, N. N., Action-Angle Variables and Their Generalization, Trans. Moscow Math. Soc., 1972, vol. 26, pp. 180­198; see also: Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 181­198 (Russian). 15. Mishchenko, A. S. and Fomenko, A. T., Generalized Liouville Metho d of Integration of Hamiltonian Systems, Funct. Anal. Appl., 1978, vol. 12, no. 2, pp. 113­121; see also: Funktsional. Anal. i Prilozhen., 1978, vol. 12, no. 2, pp. 46­56 (Russian). 16. Brailov, A. V., Complete Integrability of Some Geo desic Flows and Integrable Systems with Noncommuting Integrals, Dokl. Akad. Nauk SSSR, 1983, vol. 271, no. 2, pp. 273­276 (Russian). 17. Stekloff, W., Application du th´ ` g´ ´ alis´ de Jacobi au probl` e de Jacobi ­ Lie, C. R. Acad. Sci. eoreme ener e em Paris, 1909, vol. 148, pp. 465­468. 18. Kozlov, V. V., An Extended Hamilton ­ Jacobi Metho d, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 580­ 596. 19. Kozlov, V. V., Symmetries, Topology, and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996. 20. Kozlov, V. V., Remarks on a Lie Theorem on the Exact Integrability of Differential Equations, Differ. Equ., 2005, vol. 41, no. 4, pp. 588­590; see also: Differ. Uravn., 2005, vol. 41, no. 4, pp. 553­555 (Russian).

REGULAR AND CHAOTIC DYNAMICS

Vol. 19

No. 2

2014